Quadratic equations, sum and product of roots, and parameters – Linear and Quadratic Equations and Inequalities (Senior 4 Mathematics) | Ganzaa

Linear and Quadratic Equations and Inequalities

Lessons

Senior 4 · Mathematics

Linear and Quadratic Equations and Inequalities

5 lessons

Lesson 2 of 5

Quadratic equations, sum and product of roots, and parameters

25 min

A square and ten roots

Around the year 820, in Baghdad, Muḥammad ibn Mūsā al-Khwārizmī wrote a book whose title gave us the word algebra. One of its problems is "a square and ten roots equal thirty-nine": in today's notation $x^{2} + 10 x = 39$ . He solved it with a picture, completing a square, and found the root 3. In S3 you solved quadratic equations by factorising, by completing the square and by the formula. This lesson goes further: you will predict the nature of the roots without solving, use the sum and product of the roots, and discuss equations whose coefficients contain a parameter, a letter such as $m$ that stands for a fixed but unknown number.

1. Linear equations and inequalities with a parameter

To discuss an equation with a parameter means to solve it for every possible value of the parameter, splitting into cases wherever the method changes. For a linear equation $a x = b$ the method changes where $a = 0$ .

Worked example 1: discuss according to $m$

(a) Solve $(m - 2) x = m^{2} - 4$ . (b) Solve $(m - 1)x \gt m + 1$ .

Solution

If $m \ne 2$ : divide by $m - 2$ : $x = \frac{(m - 2)(m + 2)}{m - 2} = m + 2$ , one solution. If $m = 2$ : the equation is $0 x = 0$ , true for every real $x$ .
If $m \gt 1$ : dividing by the positive number $m - 1$ keeps the sign: $x \gt \frac{m + 1}{m - 1}$ . If $m \lt 1$ : $m - 1$ is negative, so the sign reverses: $x \lt \frac{m + 1}{m - 1}$ . If $m = 1$ : $0x \gt 2$ , which is never true, so there is no solution.

2. The quadratic formula and the discriminant

For $a x^{2} + b x + c = 0$ with $a \ne 0$ , divide by $a$ and complete the square:

x^2 + \frac{b}{a}x = -\frac{c}{a} \;\Longrightarrow\; \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} .

The left side is a square, so it can never be negative. Everything therefore depends on the sign of $b^{2} - 4 a c$ .

Definition

Discriminant of $a x^{2} + b x + c = 0$ ( $a \ne 0$ ): the number $\Delta = b^2 - 4ac$ .

$\Delta \gt 0$ : two distinct real roots $x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}$ .
$\Delta = 0$ : one double root $x = -\frac{b}{2a}$ .
$\Delta \lt 0$ : no real root.

Example: for $3 x^{2} - 5 x - 2 = 0$ , $\Delta = 25 + 24 = 49$ , so $x = \frac{5 \pm 7}{6}$ : $x = 2$ or $x = -\frac{1}{3}$ .

Tip

When $b$ is even, write $b = 2 b^{'}$ and use the reduced discriminant $\Delta' = b'^2 - ac$ (it has the same sign as $\Delta = 4\Delta'$ ); the roots are $x = \frac{-b' \pm \sqrt{\Delta'}}{a}$ . For $x^{2} - 6 x + 7 = 0$ : $\Delta' = 9 - 7 = 2$ , so $x = 3 \pm \sqrt{2}$ . Smaller numbers, fewer slips.

The discriminant counts the x-intercepts of $y = a x^{2} + b x + c$ : blue $\Delta = 16 \gt 0$ (two), green $\Delta = 0$ (one, touching), red $\Delta = -8 \lt 0$ (none).

3. Sum and product of the roots

Add and multiply the two roots given by the formula (the $\pm\sqrt{\Delta}$ terms cancel in the sum, and $(-b)^2 - \Delta = 4ac$ in the product):

x_1 + x_2 = \frac{-b + \sqrt{\Delta}}{2a} + \frac{-b - \sqrt{\Delta}}{2a} = -\frac{b}{a}, \qquad x_1 x_2 = \frac{b^2 - \Delta}{4a^2} = \frac{4ac}{4a^2} = \frac{c}{a}.

Important

Sum and product of the roots (Viète's relations). If $x_{1}$ and $x_{2}$ are the roots of $a x^{2} + b x + c = 0$ , then

S = x_1 + x_2 = -\frac{b}{a}, \qquad P = x_1 x_2 = \frac{c}{a},

and conversely two numbers with sum $S$ and product $P$ are the roots of $x^{2} - S x + P = 0$ . Useful identities: $x_{1}^{2} + x_{2}^{2} = S^{2} - 2 P$ , $(x_{1} - x_{2})^{2} = S^{2} - 4 P$ , $\frac{1}{x_1} + \frac{1}{x_2} = \frac{S}{P}$ .

Signs of the roots (when $\Delta \ge 0$ ): if $P \lt 0$ the roots have opposite signs; if $P \gt 0$ and $S \gt 0$ both are positive; if $P \gt 0$ and $S \lt 0$ both are negative.

Worked example 2: without solving

Let $\alpha$ and $\beta$ be the roots of $2 x^{2} - 6 x + 1 = 0$ . Find (a) $\alpha^2 + \beta^2$ (b) $\frac{1}{\alpha} + \frac{1}{\beta}$ (c) a quadratic equation with integer coefficients whose roots are $\alpha^2$ and $\beta^2$ .

Solution $\Delta = 36 - 8 = 28 \gt 0$ , so the roots are real. $S = \alpha + \beta = 3$ and $P = \alpha\beta = \frac{1}{2}$ .

$\alpha^2 + \beta^2 = S^2 - 2P = 9 - 1 = 8$ .
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{S}{P} = \frac{3}{\frac{1}{2}} = 6$ .
The new roots have sum $8$ and product $(\alpha\beta)^2 = \frac{1}{4}$ . The equation is $x^2 - 8x + \frac{1}{4} = 0$ , or $4 x^{2} - 32 x + 1 = 0$ .

Check: the roots are $\frac{3 \pm \sqrt{7}}{2}$ , and $\left(\frac{3 + \sqrt{7}}{2}\right)^2 + \left(\frac{3 - \sqrt{7}}{2}\right)^2 = \frac{(16 + 6\sqrt{7}) + (16 - 6\sqrt{7})}{4} = 8$ ✓.

4. Quadratic equations with a parameter

For $a (m) x^{2} + b (m) x + c (m) = 0$ , first treat the values of $m$ that make $a (m) = 0$ (the equation is then linear), then study the sign of $\Delta$ as a function of $m$ .

Worked example 3: number of roots

Discuss, according to the real parameter $m$ , the number of real roots of $(m - 1) x^{2} - 2 m x + m + 3 = 0$ .

Solution

Case $m = 1$ . The equation is $- 2 x + 4 = 0$ : one root, $x = 2$ .

Case $m \ne 1$ . Here $b = - 2 m$ is even, so $b^{'} = - m$ and

\Delta' = m^2 - (m - 1)(m + 3) = m^2 - (m^2 + 2m - 3) = 3 - 2m .

Values of $m$	Sign of $\Delta'$	Real roots
$m \lt \frac{3}{2},\ m \ne 1$	$+$	two distinct roots
$m = 1$	(linear)	one root, $x = 2$
$m = \frac{3}{2}$	$0$	one double root, $x = \frac{m}{m - 1} = 3$
$m \gt \frac{3}{2}$	$-$	none

Check $m = \frac{3}{2}$ : the equation is $\frac{1}{2}x^2 - 3x + \frac{9}{2} = 0$ , that is $x^{2} - 6 x + 9 = (x - 3)^{2} = 0$ ✓.

Warning

Do not divide by $m - 1$ or use $\Delta$ before checking $m - 1 = 0$ . When the leading coefficient is zero, the equation is not quadratic and the discriminant means nothing.

5. Two quadratic equations in one unknown

A system of two equations in the same unknown $x$ , such as $x^{2} - 5 x + 6 = 0$ and $x^{2} - 4 = 0$ , is solved by the values of $x$ that satisfy both: the common roots. Here the first gives ${2, 3}$ , the second ${- 2, 2}$ , so the system's solution is $x = 2$ . When the equations contain a parameter, subtracting them removes $x^{2}$ and leaves a linear equation for the common root.

Worked example 4: a common root

Find $m$ so that $x^{2} + m x + 2 = 0$ and $x^{2} + 2 x + m = 0$ have a common real root, and find that root.

Solution If $x$ is a common root, subtracting gives

(m - 2)x + (2 - m) = 0 \iff (m - 2)(x - 1) = 0 .

If $m = 2$ , the two equations are the same, $x^{2} + 2 x + 2 = 0$ , whose $\Delta = 4 - 8 \lt 0$ : no real root, so this case fails. If $m \ne 2$ , the common root must be $x = 1$ . Substituting in the first: $1 + m + 2 = 0$ , so $m = - 3$ .

Check: $x^{2} - 3 x + 2 = (x - 1) (x - 2)$ and $x^{2} + 2 x - 3 = (x - 1) (x + 3)$ . The common root is $1$ ✓.

The Organic Chemistry Tutor: How to find the sum and product of the roots of a quadratic equation (about 11 min). Watch how −b ÷ a and c ÷ a are read straight off the equation, and how they are checked against the roots.

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Practice

Questions 6 to 8 involve a parameter; set out every case.

Solve $2 x^{2} + 3 x - 9 = 0$ .
Solve $x^{2} - 4 x - 1 = 0$ , giving exact roots.
The roots of $x^{2} - 5 x + 2 = 0$ are $\alpha$ and $\beta$ . Find $\alpha^2 + \beta^2$ , $(\alpha - \beta)^2$ and $\alpha^3 + \beta^3$ .
Form a quadratic equation with integer coefficients whose roots are $2 + \sqrt{3}$ and $2 - \sqrt{3}$ .
Find $k$ so that $x^{2} + k x + 16 = 0$ has a double root, and give that root in each case.
Discuss according to $m$ : $(m + 1) x = m^{2} - 1$ .
Discuss according to $m$ the number of real roots of $x^{2} - 2 (m + 1) x + m^{2} + 3 = 0$ .
Find the values of $m$ for which $x^{2} - 3 x + m = 0$ and $x^{2} - 5 x + 2 m = 0$ have a common root, and give the root each time.

Show answersHide answers

$\Delta = 9 + 72 = 81$ , $x = \frac{-3 \pm 9}{4}$ : $x = \frac{3}{2}$ or $x = - 3$ .
$\Delta' = 4 + 1 = 5$ , so $x = 2 \pm \sqrt{5}$ .
$S = 5$ , $P = 2$ . $\alpha^2 + \beta^2 = 25 - 4 = 21$ ; $(\alpha - \beta)^2 = 25 - 8 = 17$ ; $\alpha^3 + \beta^3 = S^3 - 3PS = 125 - 30 = 95$ .
$S = 4$ , $P = 4 - 3 = 1$ : $x^{2} - 4 x + 1 = 0$ .
$\Delta = k^2 - 64 = 0 \Rightarrow k = \pm 8$ . For $k = 8$ the root is $- 4$ ; for $k = - 8$ it is $4$ .
$m \ne -1$ : $x = \frac{(m - 1)(m + 1)}{m + 1} = m - 1$ . $m = - 1$ : $0 x = 0$ , every real $x$ .
$\Delta' = (m + 1)^2 - (m^2 + 3) = 2m - 2$ . $m \gt 1$ : two roots; $m = 1$ : double root $x = m + 1 = 2$ ; $m \lt 1$ : no real root.
Subtracting: $2 x - m = 0$ , so $x = \frac{m}{2}$ . Then $\frac{m^2}{4} - \frac{3m}{2} + m = 0 \Rightarrow m(m - 2) = 0$ . $m = 0$ : common root $0$ . $m = 2$ : $x^{2} - 3 x + 2$ and $x^{2} - 5 x + 4$ share the root $1$ .

Quick check

Match the discriminant of a quadratic with its real roots.

Tap one on the left, then its partner on the right.

Key points

$\Delta = b^2 - 4ac$ decides the number of real roots; with $b = 2 b^{'}$ , use $\Delta' = b'^2 - ac$ .
$S = -\frac{b}{a}$ and $P = \frac{c}{a}$ ; two numbers with sum $S$ and product $P$ solve $x^{2} - S x + P = 0$ .
To discuss an equation with a parameter, treat first the values that make the leading coefficient zero, then the sign of the discriminant.
A common root of two quadratics is found by subtracting them to remove $x^{2}$ , then checking every case.

Sources and credits

Written by Ganzaa: the explanations, the graph, the worked examples, the practice items and the quiz questions. The unit follows the topic list of REB's S4 Core Mathematics (2026–2027); no REB text is reproduced.
Image: Bodleian MS. Huntington 214, text pages from al-Khwārizmī's algebra, Bodleian Libraries, Wikimedia Commons, public domain.
Video: How To Find The Sum and Product of the Roots of a Quadratic Equation, The Organic Chemistry Tutor, embedded from YouTube.

1A square and ten roots

21. Linear equations and inequalities with a parameter

32. The quadratic formula and the discriminant

43. Sum and product of the roots

54. Quadratic equations with a parameter

65. Two quadratic equations in one unknown