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Algebra 11 min read

Solving Quadratic Equations — Factorising, Formula & Completing the Square

Three methods to solve quadratic equations for GCSE Maths: factorising, the quadratic formula, and completing the square. With worked examples for AQA, Edexcel and OCR.

quadratic equationsfactorisingquadratic formulacompleting the squareHigher GCSE

What Is a Quadratic Equation?

A quadratic equation is one where the highest power of the variable is 2 — i.e., it contains an x² term. The general form is ax² + bx + c = 0. Solving it means finding the value(s) of x that make the equation true. Quadratics usually have two solutions (roots), but can have one or zero real solutions.

  • Standard form: ax² + bx + c = 0
  • Usually two solutions, sometimes one (repeated root), sometimes none (complex roots)
  • All three methods are valid — use the easiest for the given equation

Method 1: Factorising

If the quadratic factorises neatly, this is usually the quickest method. Rearrange to ax² + bx + c = 0, factorise the left side, then set each bracket equal to zero.

Worked Example

Solve x² − 5x + 6 = 0

  1. 1.Find two numbers that multiply to 6 and add to −5: −2 and −3
  2. 2.Factorise: (x − 2)(x − 3) = 0
  3. 3.Set each bracket to zero: x − 2 = 0 → x = 2; x − 3 = 0 → x = 3
Answer: x = 2 or x = 3

Method 2: The Quadratic Formula

The quadratic formula works for any quadratic, including ones that don't factorise. The formula is: x = (−b ± √(b² − 4ac)) / 2a. Identify a, b, and c from ax² + bx + c = 0, then substitute.

Worked Example

Solve 2x² + 3x − 2 = 0

  1. 1.a = 2, b = 3, c = −2
  2. 2.x = (−3 ± √(9 − 4(2)(−2))) / (2×2)
  3. 3.x = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4
  4. 4.x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2
Answer: x = 0.5 or x = −2

Method 3: Completing the Square

Completing the square rewrites x² + bx + c as (x + p)² + q. It's useful for finding the turning point of a parabola and for solving quadratics that don't factorise. For x² + bx: half the coefficient of x, square it, then add and subtract.

Worked Example

Solve x² + 6x + 7 = 0 by completing the square

  1. 1.x² + 6x + 7 = (x + 3)² − 9 + 7 = (x + 3)² − 2
  2. 2.Set equal to zero: (x + 3)² − 2 = 0
  3. 3.(x + 3)² = 2 → x + 3 = ±√2
  4. 4.x = −3 + √2 or x = −3 − √2
Answer: x = −3 + √2 or x = −3 − √2 (≈ −1.59 or −4.41)

The Discriminant — How Many Solutions?

The discriminant (b² − 4ac) tells you how many real solutions a quadratic has before you solve it. It's the expression under the square root in the quadratic formula.

  • b² − 4ac > 0: two distinct real solutions
  • b² − 4ac = 0: one repeated real solution (the curve touches the x-axis)
  • b² − 4ac < 0: no real solutions (the curve doesn't cross the x-axis)

Frequently Asked Questions

Which method should I use in an exam?

If the quadratic factorises easily, use factorising — it's fastest. If the question says "give your answer to 2 decimal places" or "use the formula", use the quadratic formula. Completing the square is usually only needed if the question explicitly asks for it or asks for the turning point.

Do I need to memorise the quadratic formula?

Yes — it's not given in the AQA, Edexcel, or OCR formula sheets. Practise writing it from memory: x = (−b ± √(b² − 4ac)) / 2a.

What if the quadratic equals a number (not zero)?

Always rearrange so the right-hand side is 0 before trying any method. For example, x² + 3x = 10 becomes x² + 3x − 10 = 0.

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