Solving Quadratic Equations | Edexcel GCSE Maths: Foundation Revision Notes 2017

Solving Quadratics by Factorising

If the equation is of the form , there is an term that is equal to a constant
- First divide both sides by $a$ and then take the square root
- For example,
  $\begin{array}{rcl} 2 x^{2} & = & 50 \\ x^{2} & = & 25 \\ x & = & \pm 5 \end{array}$

If a quadratic equation includes an $x$ term, then you will need to factorise the equation in order to solve it
Factorise the quadratic
- E.g. $x^{2} + 3 x - 4 = 0 \to (x + 4) (x - 1) = 0$
Set each bracket to zero and solve for
- Because if A × B = 0, then either A = 0 or B = 0
- For the first bracket
  $\begin{array}{rcl} x + 4 & = & 0 \\ x & = & - 4 \end{array}$
- For the second bracket
  $\begin{array}{rcl} x - 1 & = & 0 \\ x & = & 1 \end{array}$
- The two solutions are $x = - 4$ and $x = 1$
To solve a quadratic that factorises to a single bracket with a variable outside
- E.g. $x (x - 4) = 0$
- For the first bracket, it may help to think of $x$ as $(x - 0)$
  $\begin{array}{rcl} x - 0 & = & 0 \\ x & = & 0 \end{array}$
- For the second bracket
  $\begin{array}{rcl} x - 4 & = & 0 \\ x & = & 4 \end{array}$
- The two solutions are x=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} and x=4{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
  - It is a common mistake to divide both sides by $x$ at the beginning - you will lose a solution (the $x = 0$ solution)

Use a calculator to check your final solutions!
- Calculators also help you to factorise (if you're struggling with that step).

(a)

Solve $(x - 2) (x + 5) = 0$

Set the first bracket equal to zero

x – 2 = 0

Add 2 to both sides

x = 2

Set the second bracket equal to zero

x + 5 = 0

Subtract 5 from both sides

x = -5

Write both solutions together using “or”

x = 2 or x = -5

(b)

Solve $x (5 x - 1) = 0$

Do not divide both sides by x (this will lose a solution at the end)
Set the first “bracket” equal to zero

(x) = 0

Solve this equation to find x

x = 0

Set the second bracket equal to zero

5x - 1 = 0

Add 1 to both sides

5x = 1

Divide both sides by 5

x = $\frac{1}{5}$

Write both solutions together using “or”

x = 0 or x = $\frac{1}{5}$