Complete the square to solve the following quadratic equation: x2+4x=41

Question

.(1 point)
Responses

x=−2+53–√, −2−53–√
x equals negative 2 plus 5 Start Root 3 End Root comma negative 2 minus 5 Start Root 3 End Root

x=2+53–√, 2−53–√
x equals 2 plus 5 Start Root 3 End Root comma 2 minus 5 Start Root 3 End Root

x=−2+35–√, −2−35–√
x equals negative 2 plus 3 Start Root 5 End Root comma negative 2 minus 3 Start Root 5 End Root

x=2+35–√, 2−35–√

Answer 1

To solve the quadratic equation \( x^2 + 4x = 41 \) by completing the square, follow these steps:

1. Move constant to the other side:
   \[ x^2 + 4x - 41 = 0 \]

2. Complete the square:
   Take the coefficient of \( x \) (which is 4), halve it (which gives you 2), and square it (which gives you 4). Add and subtract this value on the left side:
   \[ x^2 + 4x + 4 - 4 - 41 = 0 \implies (x + 2)^2 - 45 = 0 \]

   This simplifies to:
   \[ (x + 2)^2 = 45 \]

3. Take the square root of both sides:
   \[ x + 2 = \pm \sqrt{45} \]

4. Isolate \( x \):
   \[ x = -2 \pm \sqrt{45} \]

   Note that \( \sqrt{45} = 3\sqrt{5} \), so we can simplify it: \[ x = -2 \pm 3\sqrt{5} \]

5. Final answer:
   \[ x = -2 + 3\sqrt{5}, \quad -2 - 3\sqrt{5} \]

Therefore, the correct response is: x = −2 + 3√5, −2 − 3√5.