Question

question
solve the equation for all values of x by completing the square.
3x² + 60 = 36x
answer attempt 1 out of 2
+ additional solution − no solution

Explanation:

Step1: Rearrange the equation

First, we want to get the quadratic in the form \(ax^{2}+bx = c\). Start with the given equation \(3x^{2}+60 = 36x\). Subtract \(36x\) from both sides and subtract \(60\) from both sides to get \(3x^{2}-36x=- 60\). Then, divide every term by the coefficient of \(x^{2}\) (which is 3) to simplify: \(x^{2}-12x=-20\).

Step2: Complete the square

To complete the square for the expression \(x^{2}-12x\), we take half of the coefficient of \(x\) (which is \(- 12\)), so \(\frac{-12}{2}=-6\), and then square it: \((-6)^{2} = 36\). Add this square to both sides of the equation: \(x^{2}-12x + 36=-20 + 36\).

Step3: Simplify both sides

The left side is now a perfect square trinomial, which factors as \((x - 6)^{2}\), and the right side simplifies to \(16\): \((x - 6)^{2}=16\).

Step4: Solve for x

Take the square root of both sides: \(x - 6=\pm\sqrt{16}=\pm4\). Then, solve for \(x\) by adding \(6\) to both sides. We get two solutions:

• When we take the positive square root: \(x=6 + 4=10\)
• When we take the negative square root: \(x=6-4 = 2\)

Answer:

\(x = 2\) or \(x = 10\)