The given equation in the image is:

$-4(x - 5)^2 + 5 = -4$

Step-by-step Solution:

1. Eliminate the constant on the right-hand side: Subtract $5$ from both sides of the equation: $-4(x - 5)^2 = -4 - 5$ $-4(x - 5)^2 = -9$

2. Simplify by dividing through by $-4$: $(x - 5)^2 = \frac{-9}{-4}$ $(x - 5)^2 = \frac{9}{4}$

3. Take the square root of both sides: Remember to consider both the positive and negative roots when solving quadratic equations. $x - 5 = \pm \sqrt{\frac{9}{4}}$ $x - 5 = \pm \frac{3}{2}$

4. Solve for $x$: Break it into two cases: $x - 5 = \frac{3}{2} \quad \text{or} \quad x - 5 = -\frac{3}{2}$ Add $5$ to both sides in each case: $x = 5 + \frac{3}{2} \quad \text{or} \quad x = 5 - \frac{3}{2}$ Simplify: $x = \frac{10}{2} + \frac{3}{2} = \frac{13}{2} \quad \text{or} \quad x = \frac{10}{2} - \frac{3}{2} = \frac{7}{2}$

Final Answer:

$x = \frac{13}{2} \quad \text{or} \quad x = \frac{7}{2}$

Let me know if you'd like a more detailed explanation of any step!

5 Related Questions:

1. How do you solve quadratic equations using the square root method?
2. What are the rules for simplifying square roots of fractions?
3. How do you convert improper fractions into mixed numbers?
4. Can quadratic equations always be solved by factoring?
5. Why do we consider both positive and negative roots when solving quadratic equations?

Tip:

Always check your solutions by substituting them back into the original equation to ensure they satisfy it!