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

.(1 point)
Responses

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+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 2 plus 3 Start Root 5 End Root comma 2 minus 3 Start Root 5 End Root

x=−2+53–√, −2−53–√

1 answer

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

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

  2. Complete the square: To complete the square for the left side, take half of the coefficient of \( x \) (which is 4), square it, and add it to both sides. Half of 4 is 2, and squaring it gives \( 2^2 = 4 \). \[ x^2 + 4x + 4 = 41 + 4 \] \[ (x + 2)^2 = 45 \]

  3. Solve for \( x \): Take the square root of both sides: \[ x + 2 = \pm \sqrt{45} \] Simplify \( \sqrt{45} \): \[ \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \] Now rewrite the equation: \[ x + 2 = \pm 3\sqrt{5} \] Subtract 2 from both sides: \[ x = -2 \pm 3\sqrt{5} \]

  4. Final solutions: Thus, the solutions are: \[ x = -2 + 3\sqrt{5} \quad \text{and} \quad x = -2 - 3\sqrt{5} \]

The correct response is: \[ x = -2 + 3\sqrt{5}, -2 - 3\sqrt{5} \]