Question

solve the following quadratic equation using square roots. round to the nearest hundredth if necessary: (x+16)^2/9 = 7 write answer in set notation

Answer 1

To solve the quadratic equation \(\frac{(x+16)^2}{9} = 7\) using square roots, we can follow these steps:

1. **Multiply both sides by 9** to eliminate the fraction:

\[
(x + 16)^2 = 9 \cdot 7
\]

\[
(x + 16)^2 = 63
\]

2. **Take the square root of both sides**:

\[
x + 16 = \pm \sqrt{63}
\]

3. **Simplify \(\sqrt{63}\)**:

\[
\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}
\]

So we now have:

\[
x + 16 = \pm 3\sqrt{7}
\]

4. **Solve for \(x\)**:

\[
x = -16 \pm 3\sqrt{7}
\]

Now, we will find the numerical values for these two solutions. First, we need the approximate value of \(\sqrt{7}\):

\[
\sqrt{7} \approx 2.64575 \quad (\text{rounded to five decimal places})
\]

Now calculate \(3\sqrt{7}\):

\[
3\sqrt{7} \approx 3 \times 2.64575 \approx 7.93725 \quad (\text{rounded to five decimal places})
\]

Now, we find the two solutions for \(x\):

1. **Positive root:**

\[
x_1 = -16 + 3\sqrt{7} \approx -16 + 7.93725 \approx -8.06275 \approx -8.06 \quad (\text{rounded to the nearest hundredth})
\]

2. **Negative root:**

\[
x_2 = -16 - 3\sqrt{7} \approx -16 - 7.93725 \approx -23.93725 \approx -23.94 \quad (\text{rounded to the nearest hundredth})
\]

5. **Write the solutions in set notation**:

The solutions for the equation are:

\[
\{ -8.06, -23.94 \}
\]