Completing the Square

Question

Solve \(m^2 - 4m = 32\) by “completing the square” (leave your answers in surd form)

Solution

Show solution Hide solution Fully worked — 7 steps

1. 1
   Check the arrangement of the equation

   Since both terms containing variables are on one side and the constant is on the other, proceed to step 3.

   \[ m^2 - 4m = 32 \]
2. 2
   Take ½ the coefficient of \(m\) and square it (step 3)

   \[ \left(\frac{1}{2} \times 4\right)^{2} = 4 \]
3. 3
   Add the squared value to both sides (step 4)

   \[ m^2 - 4m + 4 = 32 + 4 \]

   \[ m^2 - 4m + 4 = 36 \]
4. 4
   Factor the trinomial (step 5)

   \[ (m - 2)^{2} = 36 \]
5. 5
   Take the square root of both sides (step 6)

   \[ m - 2 = \pm 6 \]
6. 6
   Solve for \(m\) (step 7): add 2 to both sides

   \[ m = 2 \pm 6 \]

   \[ m = 2 + 6 \quad \text{or} \quad m = 2 - 6 \]
7. Final answer

   So the roots are either \(m = 8\) or \(m = -4\). That is, \(m\) is eight or \(m\) is minus four.

   \[ m = 8 \quad \text{or} \quad m = -4 \]