Complete the square to identify the two values that solve the following quadratic equation: x^2−16x−13=4

Question

why is the correct answer 17, -1
.

Answer 1

To solve the quadratic equation \( x^2 - 16x - 13 = 4 \) by completing the square, follow these steps:

1. Rearrange the equation:

   First, we want to set the equation to zero: \[ x^2 - 16x - 13 - 4 = 0 \quad \Rightarrow \quad x^2 - 16x - 17 = 0 \]

2. Move the constant term to the other side:

   We want to complete the square for the \(x\) terms: \[ x^2 - 16x = 17 \]

3. Complete the square:

   To complete the square, take half of the coefficient of \(x\) (which is \(-16\)), square it, and then add it to both sides. Half of \(-16\) is \(-8\), and squaring it gives: \[ (-8)^2 = 64 \]

   Add \(64\) to both sides: \[ x^2 - 16x + 64 = 17 + 64 \] \[ x^2 - 16x + 64 = 81 \]

4. Factor the left side:

   The left side factors to: \[ (x - 8)^2 = 81 \]

5. Take the square root of both sides:

   Now, take the square root of both sides: \[ x - 8 = \pm 9 \]

   This gives two equations: \[ x - 8 = 9 \quad \Rightarrow \quad x = 17 \] \[ x - 8 = -9 \quad \Rightarrow \quad x = -1 \]

6. The solutions:

   Thus, the two solutions to the equation \(x^2 - 16x - 13 = 4\) are: \[ \boxed{17} \quad \text{and} \quad \boxed{-1} \]

So the answer of 17 and -1 is correct.