Quadratic Equations - Revision history

Khanh: /* Geometric interpretation */

2022-01-22T23:38:07Z

Geometric interpretation

Khanh at 23:54, 10 January 2022

2022-01-10T23:54:14Z

Khanh at 23:52, 10 January 2022

2022-01-10T23:52:51Z

Khanh: /* Solving the quadratic equation */

2022-01-10T23:49:57Z

Solving the quadratic equation

Khanh: /* Quadratic factorization */

2022-01-10T23:48:03Z

Quadratic factorization

Khanh: /* Geometric interpretation */

2022-01-10T23:47:12Z

Geometric interpretation

Khanh: /* Discriminant */

2022-01-10T23:46:28Z

Discriminant

Khanh: /* Solving the quadratic equation */

2022-01-10T23:45:23Z

Solving the quadratic equation

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Khanh: /* Factoring by inspection */

2022-01-10T23:36:35Z

Factoring by inspection

Khanh: /* Solving the quadratic equation */

2022-01-10T23:35:16Z

Solving the quadratic equation

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 ===Geometric interpretation===
+[[File:Quadratic function graph key values.svg|thumb|Quadratic function graph key values|Graph of <math> y = ax^2 + bx + c</math>, where ''a'' and the discriminant <math> b^2 - 4ac</math> are positive, with <li> Roots and ''y''-intercept in <span style="color:red"> red</span></li> <li> Vertex and axis of symmetry in <span style="color:blue"> blue </span></li> <li>Focus and directrix in <span style="color:#FF1694">pink</span></li>]]
 The function ''f''(''x'') = ''ax''<sup>2</sup> + ''bx'' + ''c'' is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. As shown in Figure&nbsp;1, if {{math|''a'' &gt; 0}}, the parabola has a minimum point and opens upward. If {{math|''a'' &lt; 0}}, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The ''{{math|x}}-coordinate'' of the vertex will be located at <math>\scriptstyle x=\tfrac{-b}{2a}</math>, and the ''{{math|y}}-coordinate'' of the vertex may be found by substituting this ''{{math|x}}-value'' into the function. The ''{{math|y}}-intercept'' is located at the point {{math|(0, ''c'')}}.

← Older revision		Revision as of 23:54, 10 January 2022
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	Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.		Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.
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	==Resources==		==Resources==

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 ===Completing the square===
-[[File:Polynomialdeg2.svg|thumb|right|300px|Figure 2. For the quadratic function {{math|''y'' {{=}} ''x''<sup>2</sup> &minus; ''x'' &minus; 2}}, the points where the graph crosses the {{math|''x''}}-axis, ''x'' = −1 and ''x'' = 2, are the solutions of the quadratic equation ''x''<sup>2</sup> &minus; ''x'' &minus; 2 = 0.
+[[File:Polynomialdeg2.svg|thumb|right|300px|Figure 2. For the quadratic function ''y'' = ''x''<sup>2</sup> &minus; ''x'' &minus; 2, the points where the graph crosses the {{math|''x''}}-axis, ''x'' = −1 and ''x'' = 2, are the solutions of the quadratic equation ''x''<sup>2</sup> &minus; ''x'' &minus; 2 = 0.
 |alt=Figure 2 illustrates an x y plot of the quadratic function f of x equals x squared minus x minus 2. The x-coordinate of the points where the graph intersects the x-axis, x equals &minus;1 and x equals 2, are the solutions of the quadratic equation x squared minus x minus 2 equals zero.]]
 The process of completing the square makes use of the algebraic identity
@@ Line 145: / Line 145: @@
 ==Examples and applications==
-[[File:La Jolla Cove cliff diving - 02.jpg|thumb|The trajectory of the cliff jumper is [[parabola|parabolic]] because horizontal displacement is a linear function of time <math>x=v_x t</math>, while vertical displacement is a quadratic function of time <math>y=\tfrac{1}{2} at^2+v_y t+h</math>. As a result, the path follows quadratic equation <math>y=\tfrac{a}{2v_x^2} x^2+\tfrac{v_y}{v_x} x+h</math>, where <math>v_x</math> and <math>v_y</math> are horizontal and vertical components of the original velocity, {{math|a}} is [[Gravity of Earth|gravitational]] [[acceleration]] and {{math|h}} is original height. The {{math|a}} value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).]]
+[[File:La Jolla Cove cliff diving - 02.jpg|thumb|The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of time <math>x=v_x t</math>, while vertical displacement is a quadratic function of time <math>y=\tfrac{1}{2} at^2+v_y t+h</math>. As a result, the path follows quadratic equation <math>y=\tfrac{a}{2v_x^2} x^2+\tfrac{v_y}{v_x} x+h</math>, where <math>v_x</math> and <math>v_y</math> are horizontal and vertical components of the original velocity, {{math|a}} is gravitational acceleration and {{math|h}} is original height. The {{math|a}} value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).]]
-The [[golden ratio]] is found as the positive solution of the quadratic equation <math>x^2-x-1=0.</math>
+The golden ratio is found as the positive solution of the quadratic equation <math>x^2-x-1=0.</math>
-The equations of the [[circle]] and the other [[conic sections]]&mdash;[[ellipse]]s, [[parabola]]s, and [[hyperbola]]s&mdash;are quadratic equations in two variables.
+The equations of the circle and the other conic sections&mdash;ellipses, parabolas, and hyperbolas&mdash;are quadratic equations in two variables.
-Given the [[cosine]] or [[sine]] of an angle, finding the cosine or sine of [[Bisection#Angle bisector|the angle that is half as large]] involves solving a quadratic equation.
+Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation.
-The process of simplifying expressions involving the [[nested radical|square root of an expression involving the square root of another expression]] involves finding the two solutions of a quadratic equation.
+The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.
-[[Descartes' theorem]] states that for every four kissing (mutually tangent) circles, their [[radius|radii]] satisfy a particular quadratic equation.
+Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.
-The equation given by [[Fuss' theorem]], giving the relation among the radius of a [[bicentric quadrilateral]]'s [[inscribed circle]], the radius of its [[circumscribed circle]], and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the [[excircle]] of an [[ex-tangential quadrilateral]].
+The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral.
-[[Critical point (mathematics)|Critical points]] of a [[cubic function]] and [[inflection point]]s of a [[quartic function]] are found by solving a quadratic equation.
+Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.

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 ===Graphical solution===
-[[File:Graphical calculation of root of quadratic equation.png|240px|thumb|Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2''x''<sup>2</sup> + 4''x'' &minus; 4 = 0. Although the display shows only five significant figures of accuracy, the retrieved value of {{math|''xc''}} is 0.732050807569, accurate to twelve significant figures.]]
 [[Image:Visual.complex.root.finding.png|240px|right|thumb|A quadratic function without real root: ''y'' = (''x'' − 5)<sup>2</sup> + 9. The "3" is the imaginary part of the ''x''-intercept. The real part is the ''x''-coordinate of the vertex. Thus the roots are 5 ± 3''i''.]]

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 It follows from the quadratic formula that
 : <math>ax^2+bx+c = a \left( x - \frac{-b + \sqrt {b^2-4ac}}{2a} \right) \left( x - \frac{-b - \sqrt {b^2-4ac}}{2a} \right).</math>
-In the special case {{math|''b''<sup>2</sup> {{=}} 4''ac''}} where the quadratic has only one distinct root (''i.e.'' the discriminant is zero), the quadratic polynomial can be factored as
+In the special case ''b''<sup>2</sup> = 4''ac'' where the quadratic has only one distinct root (''i.e.'' the discriminant is zero), the quadratic polynomial can be factored as
 :<math>ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.</math>

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 ===Geometric interpretation===
-The function ''f''(''x'') = ''ax''<sup>2</sup> + ''bx'' + ''c'' is a quadratic function. The graph of any quadratic function has the same general shape, which is called a [[parabola]]. The location and size of the parabola, and how it opens, depend on the values of {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. As shown in Figure&nbsp;1, if {{math|''a'' &gt; 0}}, the parabola has a minimum point and opens upward. If {{math|''a'' &lt; 0}}, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The ''{{math|x}}-coordinate'' of the vertex will be located at <math>\scriptstyle x=\tfrac{-b}{2a}</math>, and the ''{{math|y}}-coordinate'' of the vertex may be found by substituting this ''{{math|x}}-value'' into the function. The ''{{math|y}}-intercept'' is located at the point {{math|(0, ''c'')}}.
+The function ''f''(''x'') = ''ax''<sup>2</sup> + ''bx'' + ''c'' is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. As shown in Figure&nbsp;1, if {{math|''a'' &gt; 0}}, the parabola has a minimum point and opens upward. If {{math|''a'' &lt; 0}}, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The ''{{math|x}}-coordinate'' of the vertex will be located at <math>\scriptstyle x=\tfrac{-b}{2a}</math>, and the ''{{math|y}}-coordinate'' of the vertex may be found by substituting this ''{{math|x}}-value'' into the function. The ''{{math|y}}-intercept'' is located at the point {{math|(0, ''c'')}}.
 The solutions of the quadratic equation ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0 correspond to the roots of the function ''f''(''x'') = ''ax''<sup>2</sup> + ''bx'' + ''c'', since they are the values of {{math|''x''}} for which ''f''(''x'') = 0. As shown in Figure&nbsp;2, if {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are real numbers and the domain of {{math|''f''}} is the set of real numbers, then the roots of {{math|''f''}} are exactly the {{math|''x''}}-coordinates of the points where the graph touches the {{math|''x''}}-axis. As shown in Figure&nbsp;3, if the discriminant is positive, the graph touches the {{math|''x''}}-axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the {{math|''x''}}-axis.

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 *If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots
 ::<math> -\frac{b}{2a} + i \frac{\sqrt {-\Delta}}{2a} \quad\text{and}\quad -\frac{b}{2a} - i \frac{\sqrt {-\Delta}}{2a},</math>
-:which are [[complex conjugate]]s of each other. In these expressions {{math|''i''}} is the imaginary unit.
+:which are complex conjugates of each other. In these expressions {{math|''i''}} is the imaginary unit.
 Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

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 For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form ''x''<sup>2</sup> + ''bx'' + ''c'' = 0, the sought factorization has the form {{math|(''x'' + ''q'')(''x'' + ''s'')}}, and one has to find two numbers {{math|''q''}} and {{math|''s''}} that add up to {{math| ''b''}} and whose product is {{math|''c''}} (this is sometimes called "Vieta's rule". and is related to Vieta's formulas). As an example, {{math|''x''<sup>2</sup> + 5''x'' + 6}} factors as {{math|(''x'' + 3)(''x'' + 2)}}. The more general case where {{math|''a''}} does not equal {{math|1}} can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.
-Except for special cases such as where {{math|''b'' {{=}} 0}} or ''c'' = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.
+Except for special cases such as where ''b'' = 0 or ''c'' = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.
 ===Completing the square===

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 It may be possible to express a quadratic equation ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0 as a product (''px'' + ''q'')(''rx'' + ''s'') = 0. In some cases, it is possible, by simple inspection, to determine values of ''p'', ''q'', ''r,'' and ''s'' that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if ''px'' + ''q'' = 0 or ''rx'' + ''s'' = 0. Solving these two linear equations provides the roots of the quadratic.
-For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form {{math|''x''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}}, the sought factorization has the form {{math|(''x'' + ''q'')(''x'' + ''s'')}}, and one has to find two numbers {{math|''q''}} and {{math|''s''}} that add up to {{math| ''b''}} and whose product is {{math|''c''}} (this is sometimes called "Vieta's rule". and is related to [[Vieta's formulas]]). As an example, {{math|''x''<sup>2</sup> + 5''x'' + 6}} factors as {{math|(''x'' + 3)(''x'' + 2)}}. The more general case where {{math|''a''}} does not equal {{math|1}} can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.
+For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form ''x''<sup>2</sup> + ''bx'' + ''c'' = 0, the sought factorization has the form {{math|(''x'' + ''q'')(''x'' + ''s'')}}, and one has to find two numbers {{math|''q''}} and {{math|''s''}} that add up to {{math| ''b''}} and whose product is {{math|''c''}} (this is sometimes called "Vieta's rule". and is related to Vieta's formulas). As an example, {{math|''x''<sup>2</sup> + 5''x'' + 6}} factors as {{math|(''x'' + 3)(''x'' + 2)}}. The more general case where {{math|''a''}} does not equal {{math|1}} can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.
-Except for special cases such as where {{math|''b'' {{=}} 0}} or {{math|''c'' {{=}} 0}}, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.<ref name=Washington2000/>{{rp|207}}
+Except for special cases such as where {{math|''b'' {{=}} 0}} or ''c'' = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.
 ===Completing the square===
-[[File:Polynomialdeg2.svg|thumb|right|300px|Figure 2. For the [[quadratic function]] {{math|''y'' {{=}} ''x''<sup>2</sup> &minus; ''x'' &minus; 2}}, the points where the graph crosses the {{math|''x''}}-axis, {{math|''x'' {{=}} −1}} and {{math|''x'' {{=}} 2}}, are the solutions of the quadratic equation {{math|''x''<sup>2</sup> &minus; ''x'' &minus; 2 {{=}} 0}}.
+[[File:Polynomialdeg2.svg|thumb|right|300px|Figure 2. For the quadratic function {{math|''y'' {{=}} ''x''<sup>2</sup> &minus; ''x'' &minus; 2}}, the points where the graph crosses the {{math|''x''}}-axis, ''x'' = −1 and ''x'' = 2, are the solutions of the quadratic equation ''x''<sup>2</sup> &minus; ''x'' &minus; 2 = 0.
 |alt=Figure 2 illustrates an x y plot of the quadratic function f of x equals x squared minus x minus 2. The x-coordinate of the points where the graph intersects the x-axis, x equals &minus;1 and x equals 2, are the solutions of the quadratic equation x squared minus x minus 2 equals zero.]]
 The process of completing the square makes use of the algebraic identity
 :<math>x^2+2hx+h^2 = (x+h)^2,</math>
-which represents a well-defined [[algorithm]] that can be used to solve any quadratic equation.<ref name=Washington2000/>{{rp|207}} Starting with a quadratic equation in standard form, {{math|''ax''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}}
+which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0
 #Divide each side by {{math|''a''}}, the coefficient of the squared term.
 #Subtract the constant term {{math|''c''/''a''}} from both sides.
@@ Line 38: / Line 38: @@
 #Solve each of the two linear equations.
-We illustrate use of this algorithm by solving {{math|2''x''<sup>2</sup> + 4''x'' &minus; 4 {{=}} 0}}
+We illustrate use of this algorithm by solving 2''x''<sup>2</sup> + 4''x'' &minus; 4 = 0
 :<math>1) \ x^2+2x-2=0</math>
 :<math>2) \ x^2+2x=2</math>
@@ Line 46: / Line 46: @@
 :<math>6) \ x=-1\pm\sqrt{3}</math>
-The [[plus–minus sign|plus–minus symbol "±"]] indicates that both {{math|''x'' {{=}} &minus;1 + {{radic|3}}}} and {{math|''x'' {{=}} &minus;1 &minus; {{radic|3}}}} are solutions of the quadratic equation.<ref>{{Citation|last=Sterling|first=Mary Jane|title=Algebra I For Dummies|year=2010|publisher=Wiley Publishing|isbn=978-0-470-55964-2|url=https://books.google.com/books?id=2toggaqJMzEC&q=quadratic+formula&pg=PA219|page=219}}</ref>
+The plus–minus symbol "±" indicates that both ''x'' = &minus;1 + {{radic|3}} and ''x'' = &minus;1 &minus; {{radic|3}} are solutions of the quadratic equation.
 === Quadratic formula and its derivation ===
-[[Completing the square]] can be used to [[Quadratic formula#Derivation of the formula|derive a general formula]] for solving quadratic equations, called the quadratic formula. The [[mathematical proof]] will now be briefly summarized.  It can easily be seen, by [[polynomial expansion]], that the following equation is equivalent to the quadratic equation:
+Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The mathematical proof will now be briefly summarized.  It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:
 :<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</math>
-Taking the [[square root]] of both sides, and isolating {{math|''x''}}, gives:
+Taking the square root of both sides, and isolating {{math|''x''}}, gives:
 :<math>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>
-Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as {{math|''ax''<sup>2</sup> + 2''bx'' + ''c'' {{=}} ''0''}} or {{math|''ax''<sup>2</sup> &minus; 2''bx'' + ''c'' {{=}} 0}}&nbsp;,<ref name="kahan">{{Citation |first=Willian |last=Kahan |title=On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic |url=http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf |date=November 20, 2004 |access-date=2012-12-25}}</ref> where {{math|''b''}} has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.
+Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ''ax''<sup>2</sup> + 2''bx'' + ''c'' = ''0'' or ''ax''<sup>2</sup> &minus; 2''bx'' + ''c'' = 0&nbsp;, where {{math|''b''}} has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.
-A number of [[Quadratic formula#Other derivations|alternative derivations]] can be found in the literature.  These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.
+A number of alternative derivations can be found in the literature.  These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.
-A lesser known quadratic formula, as used in [[Muller's method]] provides the same roots via the equation
+A lesser known quadratic formula, as used in Muller's method provides the same roots via the equation
 :<math>x = \frac{2c}{-b \pm \sqrt {b^2-4ac}}.</math>
-This can be deduced from the standard quadratic formula by [[Vieta's formulas]], which assert that the product of the roots is {{math|''c''/''a''}}.
+This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is {{math|''c''/''a''}}.
-One property of this form is that it yields one valid root when {{math|''a'' {{=}} 0}}, while the other root contains division by zero, because when {{math|''a'' {{=}} 0}}, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an [[indeterminate form]] {{math|0/0}} for the other root. On the other hand, when {{math|''c'' {{=}} 0}}, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form {{math|0/0}}.
+One property of this form is that it yields one valid root when ''a'' = 0, while the other root contains division by zero, because when ''a'' = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form {{math|0/0}} for the other root. On the other hand, when ''c'' = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form {{math|0/0}}.
 ===Reduced quadratic equation===
-It is sometimes convenient to reduce a quadratic equation so that its [[leading coefficient]] is one. This is done by dividing both sides by {{math|''a''}}, which is always possible since {{math|''a''}} is non-zero.  This produces the ''reduced quadratic equation'':<ref>Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. ''Concise Handbook of Mathematics and Physics'', p. 38 (CRC Press 1997)</ref>
+It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by {{math|''a''}}, which is always possible since {{math|''a''}} is non-zero.  This produces the ''reduced quadratic equation'':
 :<math>x^2+px+q=0,</math>
-where {{math|''p'' {{=}} ''b''/''a''}} and {{math|''q'' {{=}} ''c''/''a''}}. This [[monic polynomial]] equation has the same solutions as the original.
+where ''p'' = ''b''/''a'' and ''q'' = ''c''/''a''. This monic polynomial equation has the same solutions as the original.
 The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is:
@@ Line 142: / Line 142: @@
 This occurs when the roots have different [[order of magnitude]], or, equivalently, when {{math|''b''<sup>2</sup>}} and  {{math|''b''<sup>2</sup> − 4''ac''}} are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause [[loss of significance]] or [[catastrophic cancellation]] in the smaller root. To avoid this, the root that is smaller in magnitude, {{math|''r''}}, can be computed as <math>(c/a)/R</math> where {{math|''R''}} is the root that is bigger in magnitude.
-A second form of cancellation can occur between the terms {{math|''b''<sup>2</sup>}} and {{math|4''ac''}} of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.<ref name="kahan"/><ref name="Higham2002">{{Citation |first=Nicholas |last=Higham |title=Accuracy and Stability of Numerical Algorithms |edition=2nd |publisher=SIAM |year=2002 |isbn=978-0-89871-521-7 |page=10 }}</ref>
+A second form of cancellation can occur between the terms {{math|''b''<sup>2</sup>}} and {{math|4''ac''}} of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.
 ==Examples and applications==