Solve the quadratic equation by completing the square.

Question

X^2-6x- 5=0
First, choose the appropriate form and fill in the blanks with the correct numbers.
Then, solve the equation. Round your answer to the nearest hundredth.
If there is more than one solution, separate them with commas.

Form:

Solution: X = ___

Answer 1

To solve the quadratic equation \(x^2 - 6x - 5 = 0\) by completing the square, we first rearrange the equation to complete the square.

1. Move the constant term to the other side of the equation: \[ x^2 - 6x = 5 \]

2. To complete the square, take half of the coefficient of \(x\) (which is \(-6\)), square it, and add it to both sides. Half of \(-6\) is \(-3\), and squaring it gives \(9\): \[ x^2 - 6x + 9 = 5 + 9 \] \[ x^2 - 6x + 9 = 14 \]

3. Now, we can factor the left side: \[ (x - 3)^2 = 14 \]

4. Take the square root of both sides to solve for \(x\): \[ x - 3 = \pm \sqrt{14} \]

5. Solve for \(x\) by adding 3 to both sides: \[ x = 3 \pm \sqrt{14} \]

6. Now, we find the numerical values for the solutions, rounding to the nearest hundredth:

   • First solution: \[ x_1 = 3 + \sqrt{14} \approx 3 + 3.7417 \approx 6.7417 \approx 6.74 \]
   • Second solution: \[ x_2 = 3 - \sqrt{14} \approx 3 - 3.7417 \approx -0.7417 \approx -0.74 \]

Form:

The completed square form would be \((x - 3)^2 = 14\).

Solution:

The solutions are: \[ x = 6.74, -0.74 \]

So we can write: Solution: \(X = 6.74, -0.74\).