Question

Complete the square to solve the following quadratic equation: x2+14x+13=27
.(1 point)
Responses

x=7+33–√, 7−33–√
x equals 7 plus 3 Start Root 3 End Root comma 7 minus 3 Start Root 3 End Root

x=7+37–√, 7−37–√
x equals 7 plus 3 Start Root 7 End Root comma 7 minus 3 Start Root 7 End Root

x=−7+37–√, −7−37–√
x equals negative 7 plus 3 Start Root 7 End Root comma negative 7 minus 3 Start Root 7 End Root

x=−7+33–√, −7−33–√

Answer 1

To solve the quadratic equation \( x^2 + 14x + 13 = 27 \) by completing the square, we first rearrange the equation:

1. Subtract 27 from both sides:
\[
x^2 + 14x + 13 - 27 = 0
\]
This simplifies to:
\[
x^2 + 14x - 14 = 0
\]

2. Move the constant term to the right side:
\[
x^2 + 14x = 14
\]

3. To complete the square, take half of the coefficient of \( x \) (which is 14), square it, and add it to both sides. Half of 14 is 7, and squaring it gives 49:
\[
x^2 + 14x + 49 = 14 + 49
\]
So we have:
\[
(x+7)^2 = 63
\]

4. Now take the square root of both sides:
\[
x + 7 = \pm \sqrt{63}
\]

5. Subtract 7 from both sides to solve for \( x \):
\[
x = -7 \pm \sqrt{63}
\]

The expression \( \sqrt{63} \) can also be simplified:
\[
\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}
\]
6. Therefore, the solutions can be expressed as:
\[
x = -7 + 3\sqrt{7}, \quad x = -7 - 3\sqrt{7}
\]

So, the correct response is:
**x = −7 + 3√7, −7 − 3√7**.