Quadratic Equations Online Practice

Question

Complete this assessment to review what you’ve learned. It will not count toward your grade.
12 of 2012 of 20 Items

Question
Complete the square to solve the following quadratic equation: x2+4x=41
.(1 point)
Responses

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

Answer 1

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

1. Move the constant to the right side: \[ x^2 + 4x - 41 = 0 \quad \text{(but we can keep it in the original form for simplicity)} \]

2. Complete the square:

   • 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 \( 2^2 = 4 \).

   So, we update the equation: \[ x^2 + 4x + 4 = 41 + 4 \] Which simplifies to: \[ (x + 2)^2 = 45 \]

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

4. Solve for \( x \): \[ x + 2 = \pm 3\sqrt{5} \] Therefore: \[ x = -2 \pm 3\sqrt{5} \]

So the solutions are: \[ x = -2 + 3\sqrt{5}, \quad x = -2 - 3\sqrt{5} \]

From the given options, the correct response is: x = −2 + 3√5, −2 − 3√5.