Solve the equation by extracting square roots. When a solution is irrational, list both the exact solution and its approximation rounded to two decimal places. (If a solution is rational, enter NA for its approximation.)
(6x − 1)2 = 80

We are given the equation:

\[
(6x - 1)^2 = 80
\]

### Step 1: Take the square root of both sides
First, we eliminate the square by taking the square root of both sides:

\[
\sqrt{(6x - 1)^2} = \pm \sqrt{80}
\]

This simplifies to:

\[
6x - 1 = \pm \sqrt{80}
\]

### Step 2: Simplify the square root of 80
\(\sqrt{80}\) can be simplified:

\[
\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}
\]

Now, we have:

\[
6x - 1 = \pm 4\sqrt{5}
\]

### Step 3: Solve for \(x\)
Now, isolate \(x\):

\[
6x = 1 \pm 4\sqrt{5}
\]

Divide both sides by 6:

\[
x = \frac{1 \pm 4\sqrt{5}}{6}
\]

### Step 4: Find the approximation
Now, let’s calculate the approximate value of \(x\) by evaluating \(4\sqrt{5}\). First, approximate \(\sqrt{5} \approx 2.236\):

\[
4\sqrt{5} \approx 4 \times 2.236 = 8.944
\]

Now substitute this into the equation:

\[
x = \frac{1 \pm 8.944}{6}
\]

This gives us two solutions:

1. \(x_1 = \frac{1 + 8.944}{6} \approx \frac{9.944}{6} \approx 1.66\)
2. \(x_2 = \frac{1 - 8.944}{6} \approx \frac{-7.944}{6} \approx -1.32\)

### Final Answer:
- **Exact solution**: \( x = \frac{1 \pm 4\sqrt{5}}{6} \)
- **Approximate solutions**: \( x_1 \approx 1.66 \) and \( x_2 \approx -1.32 \)

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you simplify square roots involving products of numbers?
2. What are the properties of square roots when solving quadratic equations?
3. Can you solve \( (5x + 2)^2 = 45 \) using a similar method?
4. What makes a solution irrational in a square root equation?
5. How does completing the square differ from solving by square roots?

**Tip:** Always simplify square roots as much as possible before solving, to make your work easier.

Solve the equation by extracting square roots. When a solution is irrational, list both the exact solution and its approximation rounded to two decimal places. (If a solution is rational, enter NA for its approximation.) (6x − 1)^2 = 80

Learn how to solve the quadratic equation (6x − 1)^2 = 80 by extracting square roots. This step-by-step guide shows how to simplify square roots and solve for x, providing both exact and approximate solutions for irrational results. Suitable for students in Grades 8-10.

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