# Roots of quadratic equations

> Edexcel International A-Level Further Mathematics · Edexcel IAL FP1
> Source: https://www.owlsprep.com/study/edexcel-ial-further-math-fp1-roots-of-quadratic-equations/

This guide covers Edexcel IAL FP1 content on roots of quadratic equations, including sum/product formulae, symmetric expression manipulation, and construction of quadratics with transformed roots, aligned to the 2018 specification.

**Prerequisites:** [Basic quadratic equation solving and algebraic manipulation](https://www.owlsprep.com/study/edexcel-ial-maths-pure-1-quadratic-equations/); [Elementary algebraic identities (expansion of cubed binomials)](https://www.owlsprep.com/study/edexcel-ial-maths-pure-1-algebraic-expressions/)

## Learning objectives

- Recall and apply sum and product of roots formulae for quadratic equations
- Manipulate symmetric expressions in roots including squares, reciprocals and cubes using standard identities
- Construct new quadratic equations with transformed roots using computed sum and product values

## Sum and Product of Roots Formulae

For any quadratic equation of the form $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, 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.

**Sum and Product of Quadratic Roots** — Sum of roots: $\alpha + \beta = -\frac{b}{a}$. Product of roots: $\alpha\beta = \frac{c}{a}$.

*Notation:* $ax^2 + bx + c = 0$, roots $\alpha, \beta$

*Example:* For $2x^2 + 5x + 2 = 0$, $\alpha + \beta = -5/2$, $\alpha\beta = 2/2 = 1$.

**Worked example:** The quadratic equation $3x^2 - 7x + 2 = 0$ has roots $\alpha$ and $\beta$. Calculate the sum and product of its roots.

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

> **tip**
>
> Always double-check the sign of b when calculating the sum of roots: a common mistake is to omit the negative sign, leading to an incorrect sum value.

*Calculator:* allowed

## Manipulating Symmetric Expressions in Roots

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

**Key Symmetric Identities for Quadratic Roots** — You are expected to reconstruct these identities as needed: 1. $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$ 2. $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha+\beta}{\alpha\beta}$ 3. $\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)$ (this identity is explicitly required knowledge).

**Worked example:** For the quadratic $2x^2 - 4x - 1 = 0$ with roots $\alpha$ and $\beta$, find the exact value of $\alpha^3 + \beta^3$.

1. First calculate $S = \alpha+\beta$ and $P=\alpha\beta$ 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. $$= (2)^3 - 3(-\frac{1}{2})(2) = 8 + 3 = 11$$

> **Exam tip**
>
> All symmetric expressions must be evaluated exactly, so avoid using a calculator to find individual root values unless explicitly instructed to do so. Mark schemes award method marks for correct substitution into S and P even if your final calculation is wrong.

*Calculator:* allowed

## Forming Quadratics with Transformed Roots

To form a new quadratic equation with roots that are transformations of the original roots $\alpha$ and $\beta$, 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 $x^2 + 3x - 2 = 0$ has roots $\alpha$ and $\beta$. Find a quadratic equation with integer coefficients that has roots $\frac{1}{\alpha^2}$ and $\frac{1}{\beta^2}$.

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 = $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$:
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 = $\frac{1}{\alpha^2} \times \frac{1}{\beta^2}$:
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. $$4x^2 - 13x + 1 = 0$$

> **warning**
>
> Never forget the negative sign between $x^2$ and $S'x$ in the new quadratic formula. The standard form is always $x^2 - (sum)x + (product) = 0$, not $x^2 + (sum)x + (product) = 0$.

*Calculator:* allowed

## Exam Style Practice Check

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.

**Check your understanding**

1. If $\alpha+\beta = 5$ and $\alpha\beta = 2$, what is the value of $\alpha^2 + \beta^2$?

   - 21
   - 25
   - 29
   - 17

   *Why:* Correct: $\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = 25 - 4 = 21$.

2. A quadratic has new roots with sum 6 and product 8. What is the correct integer-coefficient quadratic?

   - $x^2 + 6x + 8 = 0$
   - $x^2 - 6x + 8 = 0$
   - $x^2 + 6x - 8 = 0$
   - $x^2 - 6x - 8 = 0$

   *Why:* Correct: use the standard form $x^2 - Sx + P = 0$.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Omitting the negative sign in $\alpha + \beta = -b/a$, e.g. writing $\alpha+\beta = 5$ for $x^2 +5x +3=0$.
  - Why it fails: 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: Always write the formula first before substituting values, and double-check the sign of b against the original quadratic.
- **Wrong:** Calculating individual values of $\alpha$ and $\beta$ using the quadratic formula to evaluate symmetric expressions.
  - Why it fails: This introduces unnecessary calculation errors, wastes time, and does not award method marks if you make a mistake in root calculation.
  - Correct: 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:** Writing the new quadratic as $x^2 + S'x + P' = 0$ instead of $x^2 - S'x + P' =0$.
  - Why it fails: The standard form of a quadratic with roots r1 and r2 is $(x - r1)(x - r2) = x^2 - (r1+r2)x + r1r2$, so the sum term has a negative sign.
  - Correct: Memorise the new quadratic structure explicitly as $x^2 - (sum\_of\_new\_roots)x + (product\_of\_new\_roots) = 0$.
- **Wrong:** Leaving fractional coefficients in the final quadratic when the question asks for integer coefficients.
  - Why it fails: Mark schemes deduct accuracy marks for non-integer coefficients if integer form is explicitly requested.
  - Correct: After writing the standard quadratic form, multiply all terms by the lowest common multiple of the denominators to eliminate fractions.
- **Wrong:** Using cubic or quartic Vieta formulae for quadratic root questions.
  - Why it fails: Edexcel IAL FP1 restricts this topic exclusively to quadratics, and higher-degree formulae are not required or relevant here.
  - Correct: Only use the two quadratic sum and product formulae, and reconstruct symmetric identities from basic binomial expansion rules as needed.

## Cheatsheet

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

## 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)](https://www.owlsprep.com/study/edexcel-ial-further-math-fp1-complex-numbers/)

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