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16.5 Solving Quadratic Equations Algebraically

GCSE Maths Foundation · Edexcel · Chapter 16: Quadratic Equations and Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: solving x² = k, solving by factorising, and setting up quadratic equations from problems. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Solve equations of the form x² = k, giving both solutions
  • Solve quadratic equations by factorising and setting each bracket equal to zero
  • Rearrange an equation so that one side is zero before factorising
  • Form and solve quadratic equations in context, rejecting answers that do not make sense

Key words

  • Quadratic equation: An equation with an x² term and no higher power, e.g. x² + 5x + 6 = 0.
  • Solution (root): A value of x that makes the equation true. Most quadratics have two.
  • Zero product: If A × B = 0, then A = 0 or B = 0.
  • Reject: Throw away a solution that is impossible in the context, such as a negative length.

Equations of the form x² = k

Every positive number has two square roots, one positive and one negative. Write ± ("plus or minus") to show both.

Worked example: square roots

x² = 49 gives x = ±7, because 7² = 49 and (−7)² = 49.
2x² − 8 = 0: 2x² = 8, x² = 4, x = ±2.

Answer: x = 7 or −7; x = 2 or −2

Solving by factorising

If two brackets multiply to give 0, one of them must be 0. So factorise, then set each bracket equal to zero.

Worked example: x² + 5x + 6 = 0

Factorise: two numbers that multiply to 6 and add to 5 are 2 and 3.
(x + 2)(x + 3) = 0, so x + 2 = 0 or x + 3 = 0.

Answer: x = −2 or x = −3

Worked example: x² − 7x = 0 (no number term)

Take out the common factor: x(x − 7) = 0.
x = 0 or x − 7 = 0.

Answer: x = 0 or x = 7

Key fact: Never divide both sides by x. You would lose the solution x = 0.

Rearranging first

The method only works when one side is 0. Move every term to one side first.

Worked example: x² + 2x = 15

Subtract 15: x² + 2x − 15 = 0.
Factorise: (x + 5)(x − 3) = 0.

Answer: x = −5 or x = 3

Worked example: x² − 64 = 0

Difference of two squares: (x + 8)(x − 8) = 0. (Or x² = 64, so x = ±8.)

Answer: x = −8 or x = 8

Problems leading to quadratic equations

Worked example: finding a length

A rectangle is x cm long and (x + 3) cm wide. Its area is 40 cm². Find x.
x(x + 3) = 40, so x² + 3x − 40 = 0.
(x + 8)(x − 5) = 0, so x = −8 or x = 5.
A length cannot be negative, so reject x = −8.

Answer: x = 5 (the rectangle is 5 cm by 8 cm)

Common mistakes

  • Giving only the positive answer to x² = 49. The answer is x = ±7.
  • Getting the signs wrong at the end: (x + 2)(x + 3) = 0 gives x = −2 and x = −3, not 2 and 3.
  • Factorising before making one side zero: x² + 2x = 15 cannot be solved as x(x + 2) = 15.
  • Dividing by x and losing the solution x = 0.

Revision questions

Try these, then check your answers below each one.

Question 1. Solve x² = 64. (Choose from: x = 8 or x = -8, x = 8 only, x = 32, x = 4096.)

Answer: x = 8 or x = -8

Working: 8² = 64 and (-8)² = 64, so x = ± 8.

Question 2. Leo solves x² = 5x by dividing both sides by x and gets x = 5. Has he found all the solutions?

Answer: No

Working: x(x - 5) = 0 gives x = 0 or x = 5. Dividing by x lost x = 0.

Question 3. A rectangle is x cm wide and (x + 2) cm long. Its area is 48 cm². Find x.

Answer: 6 cm

Working: x² + 2x - 48 = 0, (x + 8)(x - 6) = 0. Reject -8: x = 6.

Related lessons

Continue in the full course

This lesson is section 16.5 (EGCMF1605) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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