Roots of quadratic equations

Q: Do I get the sum and product of roots formulae in the exam formula book?

No. The formulae and are not provided in the Edexcel FP1 formula book, so you must memorise them for your exam.

Edexcel International A-Level Further Mathematics· FP1 Section 2 (2018 Spec Issue 3)· 25 min read

1. Sum and Product of Roots Formulae★★☆☆☆⏱ 5 min

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For any quadratic equation of the form $a x^{2} + b x + c = 0$ with roots $α$ and $β$ , there are two simple relationships linking the coefficients of the quadratic to the sum and product of its roots. These formulae are not provided in your exam formula book, so you must commit them to memory.

📘 Definition

Sum and Product of Quadratic Roots

$a x^{2} + b x + c = 0$ , roots $α, β$

Sum of roots: $α + β = - \frac{b}{a}$ . Product of roots: $α β = \frac{c}{a}$ .

Example:

For $2 x^{2} + 5 x + 2 = 0$ , $α + β = - 5/2$ , $α β = 2/2 = 1$ .

📐 Worked Example

The quadratic equation $3 x^{2} - 7 x + 2 = 0$ has roots $α$ and $β$ . Calculate the sum and product of its roots.

1
Identify coefficients: a = 3, b = -7, c = 2.
2
Calculate sum of roots using $α + β = - b / a$ :
3
$\alpha + \beta = -\frac{-7}{3} = \frac{7}{3}$
4
Calculate product of roots using $α β = c / a$ :
5
$\alpha\beta = \frac{2}{3}$

2. Manipulating Symmetric Expressions in Roots★★★☆☆⏱ 8 min

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Symmetric expressions in $α$ and $β$ are expressions that stay the same if you swap $α$ and $β$ . You can evaluate these without finding the individual values of $α$ and $β$ , by rewriting them in terms of $S = α + β$ and $P = α β$ using standard algebraic identities.

📘 Definition

Key Symmetric Identities for Quadratic Roots

You are expected to reconstruct these identities as needed: 1. $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ 2. $\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$ 3. $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$ (this identity is explicitly required knowledge).

📐 Worked Example

For the quadratic $2 x^{2} - 4 x - 1 = 0$ with roots $α$ and $β$ , find the exact value of $α^{3} + β^{3}$ .

1
First calculate $S = α + β$ and $P = α β$ from the given quadratic:
2
$S = -\frac{-4}{2} = 2$
3
$P = \frac{-1}{2} = -\frac{1}{2}$
4
Apply the cubic sum identity:
5
$\alpha^3 + \beta^3 = S^3 - 3PS$
6
Substitute values of S and P:
7
$= (2)^3 - 3(-\frac{1}{2})(2) = 8 + 3 = 11$

3. Forming Quadratics with Transformed Roots★★★★☆⏱ 7 min

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To form a new quadratic equation with roots that are transformations of the original roots $α$ and $β$ , you only need to calculate two values: the sum of the new roots (S') and the product of the new roots (P'). The new quadratic is then $x^{2} - S^{'} x + P^{'} = 0$ , or a scaled version with integer coefficients if required.

📐 Worked Example

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

1
First find S and P for the original quadratic:
2
$S = \alpha + \beta = -\frac{3}{1} = -3$
3
$P = \alpha\beta = \frac{-2}{1} = -2$
4
Calculate S' = sum of new roots = $\frac{1}{α ^{2}} + \frac{1}{β ^{2}}$ :
5
$S' = \frac{\beta^2 + \alpha^2}{\alpha^2\beta^2} = \frac{S^2 - 2P}{P^2}$
6
Substitute values:
7
$= \frac{(-3)^2 - 2(-2)}{(-2)^2} = \frac{9 + 4}{4} = \frac{13}{4}$
8
Calculate P' = product of new roots = $\frac{1}{α ^{2}} \times \frac{1}{β ^{2}}$ :
9
$P' = \frac{1}{(\alpha\beta)^2} = \frac{1}{P^2} = \frac{1}{4}$
10
Write the quadratic: $x^{2} - S^{'} x + P^{'} = 0$ :
11
$x^2 - \frac{13}{4}x + \frac{1}{4} = 0$
12
Multiply all terms by 4 to get integer coefficients:
13
$4x^2 - 13x + 1 = 0$

4. Exam Style Practice Check★★★★☆⏱ 5 min

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Exam questions on this topic are usually structured into parts: first asking for sum and product of roots, then evaluating a symmetric expression, then forming a new transformed quadratic. All questions require exact symbolic working, so you should show every step of your substitution into S and P.

✓ Quick check

If $α + β = 5$ and $α β = 2$ , what is the value of $α^{2} + β^{2}$ ?
- 21
- 25
- 29
- 17
Reveal answer
21 —
Correct: $α^{2} + β^{2} = (α + β)^{2} - 2 α β = 25 - 4 = 21$ .
A quadratic has new roots with sum 6 and product 8. What is the correct integer-coefficient quadratic?
- $x^{2} + 6 x + 8 = 0$
- $x^{2} - 6 x + 8 = 0$
- $x^{2} + 6 x - 8 = 0$
- $x^{2} - 6 x - 8 = 0$
Reveal answer
$x^2 - 6x + 8 = 0$ —
Correct: use the standard form $x^{2} - S x + P = 0$ .

5. Common Pitfalls

Wrong move:

Omitting the negative sign in $α + β = - b / a$ , e.g. writing $α + β = 5$ for $x^{2} + 5 x + 3 = 0$ .

Why:

The sum of roots formula explicitly uses the negative of the x coefficient, so ignoring this sign leads to all subsequent calculations being incorrect.

Correct move:

Always write the formula first before substituting values, and double-check the sign of b against the original quadratic.

Wrong move:

Calculating individual values of $α$ and $β$ using the quadratic formula to evaluate symmetric expressions.

Why:

This introduces unnecessary calculation errors, wastes time, and does not award method marks if you make a mistake in root calculation.

Correct move:

Rewrite all symmetric expressions exclusively in terms of S = α+β and P = αβ, then substitute the values calculated directly from the quadratic coefficients.

Wrong move:

Writing the new quadratic as $x^{2} + S^{'} x + P^{'} = 0$ instead of $x^{2} - S^{'} x + P^{'} = 0$ .

Why:

The standard form of a quadratic with roots r1 and r2 is $(x - r 1) (x - r 2) = x^{2} - (r 1 + r 2) x + r 1 r 2$ , so the sum term has a negative sign.

Correct move:

Memorise the new quadratic structure explicitly as $x^{2} - (s u m_o f_n e w_r oo t s) x + (p r o d u c t_o f_n e w_r oo t s) = 0$ .

Wrong move:

Leaving fractional coefficients in the final quadratic when the question asks for integer coefficients.

Why:

Mark schemes deduct accuracy marks for non-integer coefficients if integer form is explicitly requested.

Correct move:

After writing the standard quadratic form, multiply all terms by the lowest common multiple of the denominators to eliminate fractions.

Wrong move:

Using cubic or quartic Vieta formulae for quadratic root questions.

Why:

Edexcel IAL FP1 restricts this topic exclusively to quadratics, and higher-degree formulae are not required or relevant here.

Correct move:

Only use the two quadratic sum and product formulae, and reconstruct symmetric identities from basic binomial expansion rules as needed.

6. Quick Reference Cheatsheet

Concept	Formula / Rule
Sum of quadratic roots	$α + β = - \frac{b}{a}$ for $a x^{2} + b x + c = 0$
Product of quadratic roots	$α β = \frac{c}{a}$ for $a x^{2} + b x + c = 0$
Sum of squares identity	$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
Sum of reciprocals identity	$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$
Sum of cubes identity	$α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$
New quadratic formula	$x^{2} - S^{'} x + P^{'} = 0$ , where S' = sum of new roots, P' = product of new roots

7. Frequently Asked

Do I get the sum and product of roots formulae in the exam formula book?

No. The formulae $α + β = - b / a$ and $α β = c / a$ are not provided in the Edexcel FP1 formula book, so you must memorise them for your exam.

Can I use cubic root formulae for this topic?

No. Edexcel IAL FP1 restricts this topic exclusively to quadratic equations, so you will never be asked to apply sum/product rules to cubics or higher-degree polynomials in this unit.

Going deeper

syllabusEdexcel IAL Further Maths FP1 SpecificationRefer to Section 2 for roots of quadratics requirements

What's Next

Now that you have mastered roots of quadratic equations for Edexcel IAL FP1, you can build on this knowledge to tackle other core FP1 topics. Next, you should explore complex numbers, where you will use conjugate root pairs to factorise quadratics with no real roots, and apply symmetric expression manipulation to problems involving complex roots. You can also move on to coordinate systems topics, where quadratic roots are used to find intersection points between lines and conic sections. Make sure you practice past paper questions on this topic to familiarise yourself with exam phrasing and common mark scheme requirements, as this topic appears in almost every FP1 exam paper. Remember to always show full working for symmetric expression manipulation, as method marks are awarded for each step of substitution into S and P.

→Complex Numbers (FP1)