Lesson 16.4 · 30 min

Solving Quadratic Equations

Solving by factorising, forming quadratics from word problems, and trial and improvement when factorising will not work.

In this lesson you will learn to…

Solve a quadratic equation by rearranging to $= 0$ and factorising
Form a quadratic equation from a problem and interpret both solutions
Use trial and improvement to find a solution to 1 or 2 decimal places

Why this matters

Area problems, projectile paths and many geometry puzzles lead to quadratic equations. Unlike a linear equation, a quadratic usually has two answers — and deciding which one fits the real situation is part of the problem.

The five steps for word problems

Read the problem carefully.
Define the unknown(s) with a letter.
Form an equation.
Solve it.
Interpret the solutions — do they make sense?

Do not divide by $x$ . In $x^2 = 5x$ , dividing by $x$ loses the solution $x = 0$ . Rearrange to $x^2 - 5x = 0$ , factorise to $x(x - 5) = 0$ , and get both $x = 0$ and $x = 5$ .

Key rules and formulas

Zero product
$(x - 3) (x + 8) = 0 \Rightarrow x = 3 or x = - 8$
If two numbers multiply to 0, at least one of them must be 0. So set each bracket equal to 0.
Rearrange to = 0 first
$x^{2} + 5 x = 24$ must become $x^{2} + 5 x - 24 = 0$ before you factorise.
Interpret
A length or a number of people cannot be negative. Say which solution makes sense, and why.
Trial and improvement
Try values, record whether the result is too big or too small, and narrow in. To be sure of 1 d.p., test the halfway value (e.g. 3.55) between your two best guesses.

Worked examples

Worked example 1

A rectangle problem

A rectangle is 5 cm longer than it is wide. Its area is 24 cm². Find its dimensions.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Consecutive numbers

Find two consecutive numbers whose squares add up to 85.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Trial and improvement

x^{2} + 2 x = 20

has a solution between 3 and 4. Find it to 1 d.p.

Try it yourself first, then reveal the solution one step at a time.

Practice: solving quadratics

9 questions

Nine questions. Where there are two solutions, you can type them in either order.

Question 1 of 9Fill the gapsFoundation
Solve $(x - 3) (x + 8) = 0$ .
$x =$ Gap a or $x =$ Gap b
Question 2 of 9Step by step
Solve step by step.
Solve $x^{2} + 5 x = 24$ .
1. Rearrange so the right-hand side is 0. Type the left-hand side.
  Step 1 answer
Question 3 of 9Fill the gaps
Solve $x^{2} - 7 x + 12 = 0$ .
$x =$ Gap a or $x =$ Gap b
Question 4 of 9Fill the gaps
Solve $x^{2} = 5 x$ .
$x =$ Gap a or $x =$ Gap b
Question 5 of 9Type the answer
One side of a rectangle is 10 cm longer than the other. Its area is 56 cm². How long is the shorter side?
Your answercm
Question 6 of 9Type the answerChallenge
A right-angled triangle has sides $x$ , $x + 7$ and hypotenuse $x + 8$ (cm). Find $x$ .
Your answercm
Question 7 of 9Choose all that apply
Two consecutive numbers have squares that add up to 85. Select every possible pair.
Select every correct answer — there may be more than one.
5 and 6−6 and −5−7 and −66 and 7
Question 8 of 9Type the answer
One solution of $x^{2} + 3 x = 30$ is between 2 and 6. Use trial and improvement to find it to 1 d.p.
Your answer
Question 9 of 9Put in order
Put the steps for solving a quadratic word problem in order.
Put these in the right order. Drag them, or use the arrow buttons.
1. Form an equation
2. Rearrange to $= 0$ and factorise
3. Solve each bracket $= 0$
4. Define the unknown with a letter
5. Interpret: reject any solution that does not make sense

In this lesson you will learn to…

Why this matters

The five steps for word problems

Key rules and formulas

Zero product

Rearrange to = 0 first

Interpret

Trial and improvement

Worked examples

Practice: solving quadratics