Sum and product of roots | Year 11

01 / LEARN

The idea, step by step

Compare coefficients

Expanding $a (x - r) (x - s)$ gives $a x^{2} - a (r + s) x + a r s$ . Compare coefficients with $a x^{2} + b x + c$ to identify the sum and product of roots.

The sum has a minus sign. Both relationships divide by a. These formulas are useful even when the individual roots contain awkward surds.

r + s = - \frac{b}{a}, r s = \frac{c}{a}

Build or transform an equation

For sum S and product P, a monic equation with the required roots is $x^{2} - S x + P = 0$ . A nonzero constant multiple gives the same roots.

For transformed roots, find their new sum and product first. Doubling both roots doubles their sum but multiplies their product by four. Use a discriminant check if the problem also requires real roots.

A common trap

Using b/a as the root sum.

Instead: The expanded middle coefficient is -a(r+s), so the sum is -b/a.

02 / WORKED EXAMPLES

See the thinking

Example 1

Read without solving

Find the sum and product of roots of $3 x^{2} - 7 x + 2 = 0$ .

Here a=3,b=-7,c=2.
Use $r + s = - b / a$ and $r s = c / a$ .

Sum

7 / 3

and product

2 / 3

Example 2

Build the equation

Roots have sum 5 and product -6. Form a monic equation.

Use $x^{2} - S x + P = 0$ .
Substitute S=5 and P=-6.
Check factors $(x - 6) (x + 1)$ .

x^{2} - 5 x - 6 = 0

Check your understanding

Find the root product for $2 x^{2} + 9 x - 5 = 0$ .

Show answer

$- 5 / 2$

The product is c/a, including the negative sign.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find root sum/product for $x^{2} - 8 x + 10 = 0$ .
2 marks
Find root sum/product for $4 x^{2} + 3 x - 2 = 0$ .
2 marks
Form a monic equation with root sum -3 and product 2.
2 marks
One root of $x^{2} - 7 x + 12 = 0$ is 3. Find the other without full factorisation.
2 marks
Roots of $x^{2} - 4 x + 1 = 0$ are r,s. Form an equation with roots 2r,2s.
3 marks
One positive root of $x^{2} - k x + 15 = 0$ is three times the other. Find k.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find root sum/product for $x^{2} - 8 x + 10 = 0$ .
2 marks
Apply -b/a and c/a.
Sum 8; product 10.
Find root sum/product for $4 x^{2} + 3 x - 2 = 0$ .
2 marks
Divide by leading coefficient 4.
Sum -3/4; product -1/2.
Form a monic equation with root sum -3 and product 2.
2 marks
Substitute into $x^{2} - S x + P = 0$ .
$x^{2} + 3 x + 2 = 0$
One root of $x^{2} - 7 x + 12 = 0$ is 3. Find the other without full factorisation.
2 marks
The sum is 7, so the other is 4.
Product 3(4)=12 checks it.
4
Roots of $x^{2} - 4 x + 1 = 0$ are r,s. Form an equation with roots 2r,2s.
3 marks
New sum is 8 and new product is 4.
Use the monic form.
$x^{2} - 8 x + 4 = 0$
One positive root of $x^{2} - k x + 15 = 0$ is three times the other. Find k.
3 marks
Let roots t,3t, giving $3 t^{2} = 15$ .
Both are positive, so $t = \sqrt{5}$ . Add roots to find k.
$k = 4 \sqrt{5}$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

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