Solve: (x – 8)2 – 100 = 0.

The correct answer and explanation is:

Let’s solve the equation step-by-step:

Equation: (x−8)2−100=0(x – 8)^2 – 100 = 0

Step 1: Isolate the squared term

Add 100 to both sides: (x−8)2=100(x – 8)^2 = 100

Step 2: Take the square root of both sides

Recall that if a2=ba^2 = b, then a=±ba = \pm \sqrt{b}. So: x−8=±100x – 8 = \pm \sqrt{100} x−8=±10x – 8 = \pm 10

Step 3: Solve for xx in both cases

• When x−8=10x – 8 = 10,

x=10+8=18x = 10 + 8 = 18

• When x−8=−10x – 8 = -10,

x=−10+8=−2x = -10 + 8 = -2

Final solutions:

x=18orx=−2x = 18 \quad \text{or} \quad x = -2

Explanation (about 300 words):

The given equation, (x−8)2−100=0(x – 8)^2 – 100 = 0, is a quadratic equation presented in a factored or completed square form. It involves a squared binomial (x−8)2(x – 8)^2, which means the expression inside the parentheses is squared.

To solve such equations, the goal is to isolate the squared term on one side, so you can take the square root of both sides. Taking the square root is a key step because it reverses the squaring process. However, remember that when you take the square root of both sides, you must consider both the positive and negative roots because squaring either a positive or negative number yields the same result.

After isolating the squared term (x−8)2=100(x – 8)^2 = 100, we find the square root of 100, which is 10. Then, we set up two equations to reflect the positive and negative roots:

1. x−8=10x – 8 = 10
2. x−8=−10x – 8 = -10

Solving each equation gives the two values of xx that satisfy the original equation. This means the parabola y=(x−8)2−100y = (x-8)^2 – 100 crosses the x-axis at x=18x = 18 and x=−2x = -2.

This method of solving quadratic equations by taking square roots is often quicker when the equation is already in the form of a perfect square equal to a constant, rather than factoring or using the quadratic formula.

Thus, the solutions to (x−8)2−100=0(x – 8)^2 – 100 = 0 are x=18x = 18 and x=−2x = -2.