Question

solve each equation by completing the square.
$x^2 - 8x - 84 = 0$
answer
after completing the square, the equation is \boxed{} and the solution is
example: $x=3; x=5$

Explanation:

Step1: Move the constant term to the right

Given the equation \(x^2 - 8x - 84 = 0\), we first move the constant term to the right side:
\(x^2 - 8x = 84\)

Step2: Complete the square for the x-terms

To complete the square for \(x^2 - 8x\), we take half of the coefficient of \(x\) (which is \(-8\)), square it, and add it to both sides. Half of \(-8\) is \(-4\), and squaring it gives \((-4)^2 = 16\). So we add 16 to both sides:
\(x^2 - 8x + 16 = 84 + 16\)
The left side can be written as a perfect square: \((x - 4)^2 = 100\)

Step3: Solve for x

Take the square root of both sides:
\(x - 4 = \pm\sqrt{100} = \pm10\)
Then solve for \(x\) in both cases:
For \(x - 4 = 10\), we get \(x = 10 + 4 = 14\)
For \(x - 4 = -10\), we get \(x = -10 + 4 = -6\)

Answer:

After completing the square, the equation is \((x - 4)^2 = 100\) and the solution is \(x = 14\); \(x = -6\)