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:

Answer 1

To solve the quadratic equation \( x^2 - 6x - 5 = 0 \) by completing the square, we follow these steps:

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

2. To complete the square, we need to take half of the coefficient of \( x \), square it, and add it to both sides. The coefficient of \( x \) is -6.

   • Half of -6 is -3, and squaring it gives \( (-3)^2 = 9 \).

3. Add 9 to both sides: \[ x^2 - 6x + 9 = 5 + 9 \] \[ x^2 - 6x + 9 = 14 \]

4. Now, the left side can be factored as a square: \[ (x - 3)^2 = 14 \]

5. We can now take the square root of both sides: \[ x - 3 = \pm \sqrt{14} \]

6. Solving for \( x \): \[ x = 3 \pm \sqrt{14} \]

7. This gives us two solutions: \[ x = 3 + \sqrt{14} \quad \text{and} \quad x = 3 - \sqrt{14} \]

Now, we compute the numerical values of these solutions:

1. Calculate \( \sqrt{14} \): \[ \sqrt{14} \approx 3.74 \]

2. Therefore, the solutions are: \[ x_1 = 3 + 3.74 \approx 6.74 \] \[ x_2 = 3 - 3.74 \approx -0.74 \]

Finally, rounding to the nearest hundredth, we find:

Form:

The completed square form is: \[ (x - 3)^2 = 14 \]

Solution:

The solutions, rounded to the nearest hundredth, are: \[ \boxed{6.74, -0.74} \]