Quadratic Equations

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Lesson notes

Solving Quadratics by Factorising

Rearrange into ax² + bx + c = 0 with zero on one side.
Factorise the quadratic and set each bracket equal to zero.
If (x + 4)(x - 1) = 0, then x + 4 = 0 or x - 1 = 0.
For brackets with coefficients, e.g. (2x - 3)(3x + 5) = 0, solve 2x - 3 = 0 and 3x + 5 = 0.
If x is a factor, e.g. x(x - 4) = 0, solutions are x = 0 and x = 4.
Do not divide both sides by x; you will lose a solution.
Use a calculator to check factorisation: if solutions are integers or fractions, the quadratic factorises.

The Quadratic Formula

Formula: x = (-b ± √(b² - 4ac)) / (2a) for ax² + bx + c = 0 (a ≠ 0).
Read off a, b, c and substitute carefully, using brackets for negative numbers.
Simplify using a calculator or by hand; round as required (e.g. 2 d.p., 3 s.f.).
The discriminant is b² - 4ac: if > 0 two solutions, = 0 one solution, < 0 no real solutions.
If the discriminant is a perfect square, the quadratic factorises with integers.
Always show working; calculators can be used to check answers.

Completing the Square

Rewrite x² + bx as (x + p)² - p² where p = b/2.
For x² + bx + c, complete square: (x + p)² - p² + c, then simplify numbers.
If coefficient a ≠ 1, factorise a out of x² and x terms first: a[x² + (b/a)x] + c.
Then complete square inside brackets and multiply through by a.
The turning point of y = (x + p)² + q is at (-p, q); for y = a(x + p)² + q, same coordinates.
Turning point is minimum if a > 0, maximum if a < 0.
Check your answer by expanding the completed square.

Solving by Completing the Square

To solve x² + bx + c = 0, complete square: (x + p)² - p² + c = 0.
Rearrange to (x + p)² = p² - c, then take square roots: x + p = ±√(p² - c).
Solve for x: x = -p ± √(p² - c).
If a ≠ 1, divide both sides by a first (only for solving, not rewriting).
Answers are often in exact (surd) form.
Do not expand the squared bracket back out when solving.

Deciding the Quadratic Method

Use factorisation when the question says 'solve by factorising' or for simple quadratics.
Use quadratic formula when answers need a given accuracy (e.g. 2 d.p.) or when factorisation is hard.
Use completing the square when part (a) asks to complete the square and part (b) uses it to solve.
Completing the square also helps rearrange formulae with x² and x terms.
If in doubt, the quadratic formula always works.
Check solutions with a calculator: integer/fraction solutions mean factorisation works.

Slides

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Practice questions

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1.What is the standard form of a quadratic equation?
Easy
- Aax + bx + c = 0
- Bax² + bx + c = 0
- Cax² + bx + c = y
- Dax + b = 0
2.Solve x² - 4x = 0 by factorising.
Easy
- Ax = 0 or x = 4
- Bx = 0 or x = -4
- Cx = 2 or x = -2
- Dx = 4 only
3.Which of the following is the quadratic formula?
Easy
- Ax = (-b ± √(b² - 4ac)) / (2a)
- Bx = (-b ± √(b² + 4ac)) / (2a)
- Cx = (b ± √(b² - 4ac)) / (2a)
- Dx = (-b ± √(b² - 4ac)) / a
4.Use the quadratic formula to solve 3x² + 7x - 11 = 0. Give your answers correct to 2 decimal places.
Medium
- Ax = 1.08 or x = -3.41
- Bx = 1.08 or x = 3.41
- Cx = -1.08 or x = 3.41
- Dx = -1.08 or x = -3.41
5.By completing the square, solve x² + 6x + 5 = 0.
Medium
- Ax = -1 or x = -5
- Bx = 1 or x = 5
- Cx = -1 or x = 5
- Dx = 1 or x = -5
6.Write x² - 4x + 7 in the form (x - a)² + b.
Medium
- A(x - 2)² + 3
- B(x - 2)² + 7
- C(x - 4)² + 7
- D(x - 2)² + 11
7.The discriminant of 2x² - 3x + 1 = 0 is:
Easy
- A1
- B9
- C-1
- D17
8.Solve 10m² + 9m - 162 = 0.
Medium
- Am = 3.6 or m = -4.5
- Bm = -3.6 or m = 4.5
- Cm = 3.6 or m = 4.5
- Dm = -3.6 or m = -4.5

Revision notes

## Quadratic Equations - Revision Notes ### 1. Solving by Factorising - Rearrange to **ax² + bx + c = 0**. - Factorise: if (x + p)(x + q) = 0, then x = -p or x = -q. - For coefficients: (2x - 3)(3x + 5) = 0 ⇒ 2x - 3 = 0 or 3x + 5 = 0 ⇒ x = 3/2 or x = -5/3. - If x is a factor: x(x - 4) = 0 ⇒ x = 0 or x = 4. **Do not divide by x**. ### 2. Quadratic Formula - For ax² + bx + c = 0 (a ≠ 0): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] - **Discriminant** Δ = b² - 4ac: - Δ > 0: two distinct real solutions. - Δ = 0: one repeated solution. - Δ < 0: no real solutions. - Use when answers required to a given accuracy (e.g., 2 d.p.). ### 3. Completing the Square - Rewrite x² + bx as (x + p)² - p², where p = b/2. - For ax² + bx + c: factor a out of first two terms, complete square, simplify. - **Turning point**: y = a(x + p)² + q has vertex at (-p, q). - To solve: (x + p)² = q ⇒ x = -p ± √q. ### 4. Choosing a Method - **Factorising**: quick if factorable; use for simple equations. - **Quadratic formula**: always works; use for decimals/surds. - **Completing the square**: when directed or for deriving turning point. ### Key Points - Always ensure equation = 0 before solving. - Check calculator solutions to verify factorisation. - For exact surd form, simplify √(b² - 4ac) fully.

Quadratic equation: An equation of the form ax² + bx + c = 0, where a ≠ 0.
Factorisation: Writing a quadratic as a product of two linear factors, e.g., (x + p)(x + q) = 0.
Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a), used to solve any quadratic equation.
Discriminant: The value b² - 4ac under the square root in the quadratic formula; determines number of real solutions.
Completing the square: Rewriting ax² + bx + c as a(x + p)² + q to find turning point or solve.
Turning point: The vertex of a quadratic graph; minimum if a > 0, maximum if a < 0.

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