# Roots of a quadratic equation

> Edexcel International GCSE Further Pure Mathematics · 4PM1 (2016 spec)
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s2-roots-of-a-quadratic-equation/

This guide teaches you the three core methods to solve quadratic equations, plus how to use the discriminant to classify the nature of roots, aligned exactly to Edexcel IGCSE Further Pure Mathematics (4PM1) S2 requirements.

**Prerequisites:** [Factorising quadratic expressions](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s2-factorising-quadratics/); [Completing the square for quadratic expressions](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s2-completing-the-square/)

## Learning objectives

- Solve quadratic equations using 3 core methods: factorisation, completing the square, and quadratic formula
- Calculate and interpret the discriminant of a quadratic to classify the nature of its roots
- Apply these skills to standard Edexcel IGCSE FPM (4PM1) exam questions accurately

## Solving Quadratics by Factorisation

**Root of a quadratic equation** — A value of $x$ that satisfies the equation $ax^2 + bx + c = 0$, also called a solution or x-intercept of the quadratic graph.

*Example:* $x=2$ and $x=-3$ are roots of $x^2 + x - 6 = 0$

Factorisation is the fastest method for solvable quadratics, where you rewrite the quadratic as a product of two linear factors, then set each factor equal to zero to find roots. This method only works if the quadratic can be factorised over integers.

**Worked example:** Solve $2x^2 + 7x - 4 = 0$ by factorisation.

1. Find two integers that multiply to $2 \times -4 = -8$ and add to 7: 8 and -1.
2. Split the middle term and factor by grouping: $2x^2 + 8x - x -4 = 2x(x+4) -1(x+4) = (2x -1)(x +4)$
3. Set each factor equal to zero: $2x - 1 = 0 \implies x = \frac{1}{2}$; $x + 4 = 0 \implies x = -4$
4. Verify both solutions satisfy the original equation to check for arithmetic errors.

> **Exam tip:** Always check factorised roots by substituting back into the original equation if you have time to avoid careless errors.

*Calculator:* allowed

## Solving Quadratics by Completing the Square

Completing the square rewrites a quadratic in vertex form $a(x+p)^2 + q = 0$, then you rearrange to isolate $x$ to solve for roots. This method works for all real-rooted quadratics, and is the derivation of the quadratic formula.

> **warning**
>
> Never forget the $\pm$ symbol when taking square roots of both sides; missing this will cost you half the marks for the question.

**Worked example:** Solve $x^2 -6x + 2 = 0$ by completing the square, giving your answer in exact form.

1. Isolate the constant term: $x^2 -6x = -2$
2. Complete the square for the x terms: half of -6 is -3, square is 9, add to both sides: $x^2 -6x +9 = -2 +9 =7$
3. Rewrite as a squared term: $(x-3)^2 =7$
4. Take square roots of both sides (include $\pm$): $x -3 = \pm \sqrt{7}$
5. Rearrange for x: $x = 3 + \sqrt{7}$ or $x = 3 - \sqrt{7}$

*Calculator:* allowed

## Solving Quadratics using the Quadratic Formula

**Quadratic Formula** — A general formula to find roots of any quadratic equation $ax^2 +bx +c =0$ where $a \neq 0$. This formula must be memorised for your exam, as it is not provided on the formula sheet.

*Notation:* x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}

This method works for all quadratic equations, including those that cannot be factorised or require decimal answers. Pay close attention to negative values of $b$, $c$, and the discriminant when substituting values.

**Worked example:** Solve $3x^2 -5x -1 =0$ using the quadratic formula, giving answers to 2 decimal places.

1. Identify $a=3$, $b=-5$, $c=-1$ from the standard form equation.
2. Substitute into the formula: $x = \frac{-(-5) \pm \sqrt{(-5)^2 -4(3)(-1)}}{2(3)} = \frac{5 \pm \sqrt{25 +12}}{6}$
3. Simplify the discriminant: $\sqrt{37} \approx 6.0828$
4. Calculate both roots: $x = \frac{5 + 6.0828}{6} \approx 1.85$ and $x = \frac{5 - 6.0828}{6} \approx -0.18$

> **Exam tip:** When substituting negative $b$ values, write parentheses around the negative sign to avoid sign errors, e.g., $-(-5)$ instead of $--5$.

*Calculator:* allowed

## Using the Discriminant to Classify Roots

**Discriminant** — The value under the square root in the quadratic formula, which tells you the number and type of roots of a quadratic equation without solving it fully.

*Notation:* \Delta = b^2 -4ac

- If $\Delta > 0$: 2 distinct, unequal real roots
- If $\Delta = 0$: 1 repeated, equal real root
- If $\Delta < 0$: no real roots (roots are non-real complex)

**Worked example:** State the nature of the roots of $2x^2 -4x + 3 =0$.

1. Identify $a=2$, $b=-4$, $c=3$.
2. Calculate discriminant: $\Delta = (-4)^2 -4(2)(3) = 16 -24 = -8$
3. Since $\Delta = -8 < 0$, the quadratic has no real roots.

**Check your understanding**

1. What is the nature of roots of $x^2 -8x +16 =0$?

   - Two unequal real roots
   - One equal real root
   - No real roots

   *Answer:* One equal real root

   *Why:* Discriminant = $64 - 64 = 0$, so there is one repeated real root.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to include the $\pm$ symbol when taking square roots during completing the square or using the quadratic formula.
  - Why it fails: This causes you to miss one of the two roots, losing half the available marks for the question.
  - Correct: Always write the $\pm$ sign immediately after the square root symbol when solving for $x$.
- **Wrong:** Incorrectly identifying $a$, $b$, $c$ values when the quadratic is not in standard form $ax^2 +bx +c =0$.
  - Why it fails: Substituting wrong values into the quadratic formula or discriminant gives completely incorrect results.
  - Correct: Rearrange the quadratic into standard form, with all terms on one side equal to zero, before identifying $a$, $b$, $c$.
- **Wrong:** Making sign errors when substituting negative values of $b$ into the quadratic formula, e.g., writing $-5$ instead of $-(-5)$ when $b=-5$.
  - Why it fails: This flips the sign of the numerator, leading to wrong root values.
  - Correct: Wrap all negative values in parentheses when substituting into formulas to avoid sign errors.
- **Wrong:** Assuming all quadratics can be factorised, wasting time trying to factorise when the quadratic is not solvable over integers.
  - Why it fails: You lose time in the exam that could be spent on other questions.
  - Correct: If factorisation does not work after 30 seconds, switch to the quadratic formula or completing the square method.
- **Wrong:** Confusing discriminant conditions, e.g., stating $\Delta>0$ means equal roots.
  - Why it fails: This leads to incorrect classification of root type, losing all marks for that part of the question.
  - Correct: Memorise the three discriminant cases, and double-check your calculation of $\Delta$ before classifying roots.

## Cheatsheet

| Method | Use Case | Key Steps / Rules |
| --- | --- | --- |
| Factorisation | Quadratic factors easily over integers | 1. Rewrite as product of linear factors 2. Set each factor = 0 3. Solve for x |
| Completing the Square | Exact root answers, vertex form required | 1. Isolate constant term 2. Add square of half coefficient of x to both sides 3. Rearrange to solve for x, include $\pm$ |
| Quadratic Formula | All quadratics, decimal answers required | 1. Recall formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ 2. Substitute a,b,c correctly 3. Simplify to required precision |
| Discriminant | Classify root type without solving | 1. $\Delta = b^2 -4ac$ 2. $\Delta>0$: 2 unequal real roots 3. $\Delta=0$: 1 equal real root 4. $\Delta<0$: No real roots |

## What's next

Now that you have mastered solving quadratic equations and classifying their roots, you are ready to move on to more advanced quadratic function topics in the Edexcel IGCSE Further Pure Mathematics syllabus. The next core sub-topic covers symmetric functions of roots, where you will learn to use the sum and product of roots without solving the full quadratic equation. You can also practice applying these root-solving skills to coordinate geometry problems involving quadratic graphs, which appear frequently in exam papers. Make sure to practice past paper questions to reinforce your understanding of all three solution methods and discriminant classification, as these skills are foundational for almost all other algebra topics in the course.

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