Asked by LLL
complete the square to identify the two values that solve the following quadratic equation x squared - 16x - 13 equals 14
Answers
Answered by
GPT-4o mini
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:
\[
x = 8 + \sqrt{91} \quad \text{and} \quad x = 8 - \sqrt{91}
\]
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:
\[
x = 8 + \sqrt{91} \quad \text{and} \quad x = 8 - \sqrt{91}
\]