To solve the equation [tex]\(-2(x-2)^2 + 5 = 0\)[/tex], let's go through a step-by-step process:
1. Isolate the quadratic term:
To isolate the quadratic term, start by moving the constant term to the other side of the equation.
[tex]\[
-2(x-2)^2 = -5
\][/tex]
2. Divide by the coefficient of the quadratic term:
To simplify the equation, divide both sides by -2.
[tex]\[
(x-2)^2 = \frac{5}{2}
\][/tex]
[tex]\[
(x-2)^2 = 2.5
\][/tex]
3. Take the square root of both sides:
Solve for [tex]\(x - 2\)[/tex] by taking the square root of both sides.
[tex]\[
x - 2 = \pm \sqrt{2.5}
\][/tex]
Recall that taking the square root introduces both positive and negative solutions.
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], add 2 to both positive and negative roots.
[tex]\[
x = 2 + \sqrt{2.5}
\][/tex]
[tex]\[
x = 2 - \sqrt{2.5}
\][/tex]
5. Calculate the numerical values:
We'll calculate each value separately.
[tex]\[
\sqrt{2.5} \approx 1.58
\][/tex]
So,
[tex]\[
x_1 = 2 + 1.58 = 3.58
\][/tex]
[tex]\[
x_2 = 2 - 1.58 = 0.42
\][/tex]
Thus, the solutions to the equation [tex]\(-2(x-2)^2 + 5 = 0\)[/tex], rounded to the nearest hundredth, are:
[tex]\[
x = 3.58 \quad \text{and} \quad x = 0.42
\][/tex]
From the given options, the correct answer is:
[tex]\[
\boxed{x = 3.58 \quad \text{and} \quad x = 0.42}
\][/tex]
