complete the square to identify the two values that solve the following quadratic equation x squared - 16x - 13 equals 14

Answer 1

To solve the quadratic equation \( x^2 - 16x - 13 = 14 \) by completing the square, first, we need to rearrange the equation into a standard form:

1. Subtract 14 from both sides:

   \[ x^2 - 16x - 13 - 14 = 0 \]

   This simplifies to:

   \[ x^2 - 16x - 27 = 0 \]

2. Now, let's complete the square. We want to express the quadratic in the form \( (x - p)^2 = q \).

3. First, take the coefficient of \( x \), which is \(-16\), halve it, and then square it:

   \[ \left( \frac{-16}{2} \right)^2 = (-8)^2 = 64 \]

4. Next, add and subtract \( 64 \) to the left-hand side of the equation:

   \[ x^2 - 16x + 64 - 64 - 27 = 0 \]

   This can be rearranged to:

   \[ (x - 8)^2 - 91 = 0 \]

5. Now isolate the squared term:

   \[ (x - 8)^2 = 91 \]

6. Take the square root of both sides:

   \[ x - 8 = \pm \sqrt{91} \]

7. Finally, solve for \( x \):

   \[ x = 8 \pm \sqrt{91} \]

Thus, the two values that solve the equation are:

\[ x = 8 + \sqrt{91} \quad \text{and} \quad x = 8 - \sqrt{91} \]