To solve the quadratic equation $4x^2 + 5x - 8 = 0$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side

First, rewrite the equation as:

$4x^2 + 5x = 8$

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

Since the coefficient of $x^2$ is 4, divide the whole equation by 4 to simplify:

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

Step 3: Complete the square

To complete the square, take half of the coefficient of $x$, square it, and add it to both sides of the equation.

The coefficient of $x$ is $\frac{5}{4}$, so half of that is $\frac{5}{8}$. Squaring it gives:

$\left(\frac{5}{8}\right)^2 = \frac{25}{64}$

Now, add $\frac{25}{64}$ to both sides:

$x^2 + \frac{5}{4}x + \frac{25}{64} = 2 + \frac{25}{64}$

Step 4: Simplify the equation

On the left-hand side, the expression is now a perfect square trinomial. The right-hand side can be simplified:

$\left(x + \frac{5}{8}\right)^2 = \frac{128}{64} + \frac{25}{64} = \frac{153}{64}$

Step 5: Solve for $x$

Take the square root of both sides:

$x + \frac{5}{8} = \pm \sqrt{\frac{153}{64}} = \pm \frac{\sqrt{153}}{8}$

Step 6: Isolate $x$

Now, subtract $\frac{5}{8}$ from both sides:

$x = -\frac{5}{8} \pm \frac{\sqrt{153}}{8}$

Thus, the two real solutions are:

$x = \frac{-5 + \sqrt{153}}{8} \quad \text{or} \quad x = \frac{-5 - \sqrt{153}}{8}$

These are the real solutions of the equation.

Would you like more details on any step?

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Here are some related questions to explore:

1. How can completing the square help in solving quadratic equations with complex solutions?
2. What are the benefits of completing the square versus using the quadratic formula?
3. How do you identify perfect square trinomials in different quadratic expressions?
4. Can the completing-the-square method be applied to non-quadratic polynomials?
5. How can we use completing the square to derive the quadratic formula?

Tip: When completing the square, always remember to adjust for any leading coefficients of $x^2$ by dividing them out first.