Question

4x² + 16x = 36; 4x² + 16x + 256 = 36 + 256; (2x + 4)² + 256 = 36 + 256

Explanation:

Step1: Identify the equation

The equation given is \(4x^{2}+32x = 36\). First, we can factor out the greatest common factor from the left - hand side. The GCF of \(4x^{2}\) and \(32x\) is 4. So, we divide each term by 4:
\(\frac{4x^{2}}{4}+\frac{32x}{4}=\frac{36}{4}\)
Which simplifies to \(x^{2}+8x = 9\).

Step2: Complete the square

To complete the square for the quadratic expression \(x^{2}+8x\), we take half of the coefficient of \(x\), which is \(\frac{8}{2}=4\), and then square it: \(4^{2}=16\). We add and subtract 16 on the left - hand side (or just add 16 to both sides to keep the equation balanced).
\(x^{2}+8x + 16=9 + 16\)

Step3: Rewrite as a perfect square

The left - hand side \(x^{2}+8x + 16\) is a perfect square trinomial, which can be written as \((x + 4)^{2}\). So, \((x + 4)^{2}=25\).

Step4: Solve for x

Take the square root of both sides: \(x + 4=\pm\sqrt{25}=\pm5\).
Case 1: When \(x + 4 = 5\), then \(x=5 - 4=1\).
Case 2: When \(x + 4=-5\), then \(x=-5 - 4=-9\).

Answer:

The solutions of the equation \(4x^{2}+32x = 36\) are \(x = 1\) or \(x=-9\)