Complete the square to solve the following quadratic equation: x2+14x+13=27

Question

.(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+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+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–√

Answer 1

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

1. Rearrange the equation: Move all terms to one side of the equation:

   \[ x^2 + 14x + 13 - 27 = 0 \]

   Simplifying gives:

   \[ x^2 + 14x - 14 = 0 \]

2. Complete the square: We need to manipulate the quadratic expression \( x^2 + 14x \).

   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 \( 49 \).

   Therefore, we rewrite the expression as follows:

   \[ x^2 + 14x + 49 - 49 - 14 = 0 \]

   This can be reorganized as:

   \[ (x + 7)^2 - 63 = 0 \]

3. Solve for \( x \): Now set the equation to zero and solve:

   \[ (x + 7)^2 = 63 \]

   Taking the square root of both sides gives:

   \[ x + 7 = \pm \sqrt{63} \]

   Finally, solve for \( x \):

   \[ x = -7 \pm \sqrt{63} \]

4. Simplify \( \sqrt{63} \): We can simplify \( \sqrt{63} \) to \( 3\sqrt{7} \):

   \[ x = -7 \pm 3\sqrt{7} \]

Therefore, the solutions are:

\[ x = -7 + 3\sqrt{7}, \quad x = -7 - 3\sqrt{7} \]

So the correct response is:

x=−7+3√7, −7−3√7