Question

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

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–√
x equals negative 7 plus 3 Start Root 3 End Root comma negative 7 minus 3 Start Root 3 End Root

x=7+33–√, 7−33–√

Answer 1

To solve the quadratic equation \( x^2 + 14x + 13 = 27 \) by completing the square, we can follow these steps:

1. **Rearrange the equation**:
\[
x^2 + 14x + 13 - 27 = 0
\]
Simplifying further,
\[
x^2 + 14x - 14 = 0
\]

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

3. **Complete the square**:
To complete the square, take half of the coefficient of \( x \) (which is 14), square it, and add to both sides:
\[
\left(\frac{14}{2}\right)^2 = 7^2 = 49
\]
So we add 49 to both sides:
\[
x^2 + 14x + 49 = 14 + 49
\]
This simplifies to:
\[
x^2 + 14x + 49 = 63
\]

4. **Factor the left side**:
The left side can be factored as:
\[
(x + 7)^2 = 63
\]

5. **Take the square root of both sides**:
\[
x + 7 = \pm \sqrt{63}
\]

6. **Solve for \( x \)**:
\[
x = -7 \pm \sqrt{63}
\]

Now, we can simplify \( \sqrt{63} \):
\[
\sqrt{63} = 3\sqrt{7}
\]

So the solutions for \( x \) are:
\[
x = -7 + 3\sqrt{7}, \quad x = -7 - 3\sqrt{7}
\]

Thus, the answer is:
\[
\text{x equals } -7 + 3\sqrt{7} \text{, } -7 - 3\sqrt{7}
\]

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