What is the standard form of a quadratic equation?

ax² + bx + c = 0, where a ≠ 0.

What is the quadratic formula?

x = (−b ± √(b² − 4ac))/(2a)

When solving by factorisation, what do you do after factorising?

Set each bracket equal to 0 and solve for x.

Which method always works for any quadratic?

Completing the square and the quadratic formula both work for any quadratic. Factorisation only works when the roots are rational.

x² = 9 → x = 3 or x = −3. (Difference of two squares: (x−3)(x+3) = 0)

x(x + 4) = 0 → x = 0 or x = −4

What is the first step when given an unusual quadratic equation?

Rearrange it into the form ax² + bx + c = 0.

Which method is best when you also need the turning point?

Completing the square, because it gives the form a(x+p)²+q with vertex (−p, q).

What is the discriminant in the quadratic formula?

b² − 4ac, the expression under the square root. It determines the number of real roots.

Solving Quadratic Equations | Pearson Edexcel International AS Maths

Solving Quadratic Equations

Summary

Quadratic equations in the form ax² + bx + c = 0 can be solved by three main methods: factorisation (fast when possible), completing the square (always works, useful for turning points), and the quadratic formula (universal). Rearrange unusual equations into standard form first.

1. Standard Form

We solve quadratic equations when they are in the form:

$ax^2 + bx + c = 0$

If given an unusual equation, rearrange it into this form first. Identify (a), (b), and (c) (where (a \neq 0)).

2. Three Methods to Solve Quadratics

3. Solving by Factorisation

Example: Solve (x^2 - 5x + 6 = 0)

Factorise: ((x - 2)(x - 3) = 0)

Set each bracket to zero:

(x - 2 = 0) → (x = 2)
(x - 3 = 0) → (x = 3)

Solutions: (x = 2) or (x = 3)

If the numbers are simple, try factorising first. It will not work for all quadratics (e.g. when the discriminant is not a perfect square).

4. Solving by Quadratic Formula

5. Solving by Completing the Square

Example: Solve (x^2 + 6x + 5 = 0)

Complete the square: (x^2 + 6x = (x + 3)^2 - 9)

((x + 3)^2 - 9 + 5 = 0) → ((x + 3)^2 = 4)

(x + 3 = \pm 2) → (x = -1) or (x = -5)

Completing the square always works and is useful when you also need the turning point.

6. Using a Calculator

7. Examiner Tips

Solving Quadratic Equations

Summary

1. Standard Form

We solve quadratic equations when they are in the form:

$ax^2 + bx + c = 0$

If given an unusual equation, rearrange it into this form first. Identify (a), (b), and (c) (where (a \neq 0)).

2. Three Methods to Solve Quadratics

1. Factorisation – Fast when the numbers are simple, but does not work for all quadratics. Once factorised, set each bracket to 0 and solve.

2. Completing the square – Works for any quadratic. Also helps find the turning point. Write in the form (a(x + p)^2 + q = 0) and solve.

3. Quadratic formula – Works for any quadratic. Uses the coefficients (a), (b), and (c):

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression (b^2 - 4ac) is the discriminant.