To solve the quadratic equation $2x^2 - 8x - 5 = 0$ by completing the square, follow these steps:

Step 1: Divide through by the coefficient of $x^2$

We start by dividing the entire equation by 2, so the coefficient of $x^2$ becomes 1:

$x^2 - 4x - \frac{5}{2} = 0$

Step 2: Move the constant to the other side

Next, we move the constant term to the right-hand side:

$x^2 - 4x = \frac{5}{2}$

Step 3: Complete the square

To complete the square, we need to add $\left(\frac{b}{2}\right)^2$ to both sides, where $b = -4$ is the coefficient of $x$.

$\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4$

Add 4 to both sides of the equation:

$x^2 - 4x + 4 = \frac{5}{2} + 4$

$x^2 - 4x + 4 = \frac{5}{2} + \frac{8}{2} = \frac{13}{2}$

Step 4: Rewrite the left side as a square

The left-hand side is now a perfect square trinomial, so we can factor it:

$(x - 2)^2 = \frac{13}{2}$

Step 5: Solve for $x$

Now, take the square root of both sides:

$x - 2 = \pm \sqrt{\frac{13}{2}} = \pm \frac{\sqrt{13}}{\sqrt{2}} = \pm \frac{\sqrt{26}}{2}$

Step 6: Isolate $x$

Finally, solve for $x$:

$x = 2 \pm \frac{\sqrt{26}}{2}$

Thus, the solutions are:

$x = 2 + \frac{\sqrt{26}}{2} \quad \text{or} \quad x = 2 - \frac{\sqrt{26}}{2}$

These are the solutions in exact form.

Would you like further details on any of the steps?

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Related questions:

1. How does completing the square relate to the quadratic formula?
2. Can all quadratic equations be solved by completing the square?
3. What are the advantages of completing the square over factoring?
4. Can you apply completing the square to quadratic equations with complex solutions?
5. How does completing the square help in solving optimization problems?

Tip:

Completing the square is useful for rewriting quadratics in vertex form, which is helpful for analyzing parabolas and solving geometry-related problems.