To solve the quadratic equation $x^2 + 7x - \frac{23}{4} = 0$ by completing the square, let's walk through the steps.

Step 1: Rewrite the equation in the form of $x^2 + bx = c$

We already have: $x^2 + 7x = \frac{23}{4}$

Step 2: 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 7, so: $\left( \frac{7}{2} \right)^2 = \frac{49}{4}$

Add $\frac{49}{4}$ to both sides: $x^2 + 7x + \frac{49}{4} = \frac{23}{4} + \frac{49}{4}$ $\left( x + \frac{7}{2} \right)^2 = \frac{72}{4} = 18$

Step 3: Solve for $x$

Now, take the square root of both sides: $x + \frac{7}{2} = \pm \sqrt{18}$

Simplifying $\sqrt{18}$: $x + \frac{7}{2} = \pm 3\sqrt{2}$

Subtract $\frac{7}{2}$ from both sides: $x = -\frac{7}{2} \pm 3\sqrt{2}$

Thus, the solutions are: $x = -\frac{7}{2} + 3\sqrt{2} \quad \text{or} \quad x = -\frac{7}{2} - 3\sqrt{2}$

This is in the required form $x = a \pm b\sqrt{c}$, where:

• $a = -\frac{7}{2}$,
• $b = 3$,
• $c = 2$.

Would you like any further details or clarifications?

Here are 5 related questions:

1. How do you factor a quadratic equation?
2. Can completing the square be used for any quadratic equation?
3. What is the relationship between completing the square and the quadratic formula?
4. How do you simplify square roots like $\sqrt{18}$?
5. When is completing the square the best method for solving quadratics?

Tip: Completing the square is a great technique to derive the quadratic formula, as it works for any quadratic equation!