Functions of the roots of a quadratic equation

Q: Are the sum and product of roots formulas given in the exam?

No, the sum () and product () identities are not provided on the 4PM1 formula sheet, so you must memorize them.

Edexcel International GCSE Further Pure Mathematics· 4PM1 2016 Spec Section 2C· 15 min read

1. Sum and Product of Quadratic Roots Identities★★☆☆☆⏱ 4 min

✓ Calculator OK

For any quadratic equation of the form $a x^{2} + b x + c = 0$ with roots $α$ and $β$ , two core identities link the coefficients of the quadratic to its roots. These formulas are not provided in your exam formula sheet, so you must memorize them.

📘 Definition

Sum and Product of Quadratic Roots

If $a x^{2} + b x + c = 0$ has roots $α, β$ :
Sum of roots: $α + β = - \frac{b}{a}$
Product of roots: $α β = \frac{c}{a}$

Example:

For $2 x^{2} + 5 x - 3 = 0$ , $α + β = - 5/2$ , $α β = - 3/2$

📐 Worked Example

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

1
Step 1: Identify coefficients: $a = 3$ , $b = - 7$ , $c = 2$
2
Step 2: Apply sum of roots formula:
3
$\alpha + \beta = -\frac{b}{a} = -\left(\frac{-7}{3}\right) = \frac{7}{3}$
4
Step 3: Apply product of roots formula:
5
$\alpha\beta = \frac{c}{a} = \frac{2}{3}$

2. Evaluating Symmetric Functions of Roots★★★☆☆⏱ 6 min

✓ Calculator OK

Symmetric functions of roots are expressions where swapping $α$ and $β$ leaves the value unchanged. You do not need to calculate individual roots to evaluate these: all can be rewritten using only the sum and product of roots identities.

$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$
$α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$
$(α - β)^{2} = (α + β)^{2} - 4 α β$

📐 Worked Example

Given $x^{2} - 4 x + 1 = 0$ has roots $α, β$ , find the value of $α^{3} + β^{3}$ .

1
Step 1: Calculate sum and product of original roots:
2
$\alpha + \beta = 4, \quad \alpha\beta = 1$
3
Step 2: Substitute into the sum of cubes identity:
4
$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$
5
$= 4^3 - 3(1)(4) = 64 - 12 = 52$

3. Forming New Quadratics with Transformed Roots★★★☆☆⏱ 5 min

✓ Calculator OK

Once you can calculate the sum and product of a transformed set of roots, you can write the full quadratic equation for those roots using a standard form.

📘 Definition

Quadratic Equation from Roots

If a quadratic has roots $p$ and $q$ , its standard form is:
$x^{2} - (p + q) x + pq = 0$
Multiply through by a constant to clear fractions if integer coefficients are required.

Example:

A quadratic with roots 3 and 5 has equation $x^{2} - 8 x + 15 = 0$

📐 Worked Example

The quadratic $2 x^{2} + 3 x - 2 = 0$ has roots $α, β$ . Form a quadratic with integer coefficients that has roots $2 α + 1$ and $2 β + 1$ .

1
Step 1: Calculate original sum and product of roots:
2
$\alpha + \beta = -\frac{3}{2}, \quad \alpha\beta = -1$
3
Step 2: Calculate sum of new roots $S$ :
4
$S = (2\alpha +1) + (2\beta +1) = 2(\alpha + \beta) + 2 = 2\left(-\frac{3}{2}\right) + 2 = -1$
5
Step 3: Calculate product of new roots $P$ :
6
$P = (2\alpha +1)(2\beta +1) = 4\alpha\beta + 2(\alpha + \beta) +1 = 4(-1) + 2\left(-\frac{3}{2}\right) +1 = -6$
7
Step 4: Substitute into standard quadratic form:
8
$x^2 - Sx + P = x^2 +x -6 = 0$

4. Common Pitfalls

Wrong move:

Using $α + β = b / a$ instead of $- b / a$

Why:

Forgetting the leading negative sign in the sum of roots identity invalidates all subsequent calculations

Correct move:

Explicitly memorize the identity with the negative sign, and double-check the sign of $b$ when substituting values

Wrong move:

Calculating individual $α$ and $β$ values to evaluate symmetric functions

Why:

This is time-consuming and increases risk of arithmetic errors, especially with irrational roots

Correct move:

Always rewrite symmetric functions using only the sum and product of roots identities

Wrong move:

Writing new quadratics as $x^{2} + (s u m) x + p r o d u c t = 0$

Why:

Mixing up the sign of the $x$ term when forming a quadratic from roots leads to an incorrect final equation

Correct move:

Recall the standard form: subtract the sum of roots, add the product of roots

Wrong move:

Leaving fractions in the final quadratic when integer coefficients are required

Why:

Exam questions often explicitly require integer coefficients, so un-cleared fractions lose marks even if sum/product are correct

Correct move:

Multiply all terms by the lowest common denominator of any fractions to get integer coefficients

Wrong move:

Expanding $(α - β)^{2}$ as $α^{2} - β^{2}$

Why:

Confusing the square of a difference with the difference of squares leads to an incorrect symmetric identity

Correct move:

Use the pre-derived identity $(α - β)^{2} = (α + β)^{2} - 4 α β$ to avoid expansion errors

5. Quick Reference Cheatsheet

Identity Name	Formula
Sum of roots	$α + β = - \frac{b}{a}$
Product of roots	$α β = \frac{c}{a}$
Sum of squares	$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
Sum of reciprocals	$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$
Sum of cubes	$α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$
Square of difference	$(α - β)^{2} = (α + β)^{2} - 4 α β$
Quadratic from roots	$x^{2} - (sum of roots) x + (product of roots) = 0$

6. Frequently Asked

Are the sum and product of roots formulas given in the exam?

No, the sum ( $α + β = - b / a$ ) and product ( $α β = c / a$ ) identities are not provided on the 4PM1 formula sheet, so you must memorize them.

Do I need to calculate individual roots to evaluate symmetric functions?

No, all symmetric functions can be rewritten using only the sum and product of roots, which saves time and reduces calculation errors.

Going deeper

syllabusEdexcel 4PM1 Official SpecificationReference for section S2 content

What's Next

Now that you have mastered functions of quadratic roots, you are ready to move on to more advanced quadratic function topics in Edexcel IGCSE Further Pure Math 4PM1. This skill is foundational for many algebra topics you will encounter if you progress to A Level Further Mathematics, but for your 4PM1 exam, you should practice applying these identities to a range of exam-style questions, including multi-step problems that combine this topic with other algebra skills. Make sure you memorize all the core identities covered, as they are not provided on your formula sheet, and practice forming quadratics with different transformed root types to build confidence for your exam.