(a) Express \( x^{2}-4 x-32 \) in the form \( (x+a)^{2}+b \) (b) Hence, solve the equation \( x^{2}-4 x-32=0 \)

**(a) Express \( x^{2} - 4x - 32 \) in the form \( (x + a)^{2} + b \)** To express the quadratic expression \( x^{2} - 4x - 32 \) in the form \( (x + a)^{2} + b \), we'll complete the square. 1. **Start with the original expression:** \[ x^{2} - 4x - 32 \] 2. **Focus on the quadratic and linear terms:** \[ x^{2} - 4x \] 3. **Complete the square:** - Take half of the coefficient of \( x \), which is \( -4 \), so half is \( -2 \). - Square this value: \( (-2)^2 = 4 \). 4. **Add and subtract this square inside the expression:** \[ x^{2} - 4x + 4 - 4 - 32 = (x^{2} - 4x + 4) - 36 \] 5. **Rewrite the perfect square trinomial and simplify:** \[ (x - 2)^2 - 36 \] So, the expression in the desired form is: \[ x^{2} - 4x - 32 = (x - 2)^2 - 36 \] **(b) Solve the equation \( x^{2} - 4x - 32 = 0 \)** Using the completed square form from part (a): \[ (x - 2)^2 - 36 = 0 \] 1. **Isolate the squared term:** \[ (x - 2)^2 = 36 \] 2. **Take the square root of both sides:** \[ x - 2 = \pm 6 \] 3. **Solve for \( x \):** - When \( x - 2 = 6 \): \[ x = 6 + 2 = 8 \] - When \( x - 2 = -6 \): \[ x = -6 + 2 = -4 \] **Solution:** \[ x = 8 \quad \text{or} \quad x = -4 \]