Solved College Algebra Worksheet Name: Date: Use the square ... - Free Printable

Let’s solve each problem one by one using the square root property.
The square root property says:
If \( x^2 = a \), then \( x = \sqrt{a} \) or \( x = -\sqrt{a} \).
We can write that as \( x = \pm \sqrt{a} \).

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Problem 1: \( x^2 = 27 \)

Take the square root of both sides:

\( x = \pm \sqrt{27} \)

Simplify \( \sqrt{27} \):
\( \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \)

So,
\( x = \pm 3\sqrt{3} \)

✔ Final Answer for #1: \( x = 3\sqrt{3} \) or \( x = -3\sqrt{3} \)

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Problem 2: \( (x + 3)^2 = 64 \)

Take the square root of both sides:

\( x + 3 = \pm \sqrt{64} \)

\( \sqrt{64} = 8 \), so:

\( x + 3 = 8 \) or \( x + 3 = -8 \)

Now solve each:

- If \( x + 3 = 8 \), subtract 3: \( x = 5 \)
- If \( x + 3 = -8 \), subtract 3: \( x = -11 \)

✔ Final Answer for #2: \( x = 5 \) or \( x = -11 \)

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Problem 3: \( (4x + 3)^2 = 24 \)

Take the square root of both sides:

\( 4x + 3 = \pm \sqrt{24} \)

Simplify \( \sqrt{24} \):
\( \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \)

So:

\( 4x + 3 = 2\sqrt{6} \) or \( 4x + 3 = -2\sqrt{6} \)

Now solve each equation for \( x \):

First equation:
\( 4x + 3 = 2\sqrt{6} \)
Subtract 3: \( 4x = 2\sqrt{6} - 3 \)
Divide by 4: \( x = \frac{2\sqrt{6} - 3}{4} \)

Second equation:
\( 4x + 3 = -2\sqrt{6} \)
Subtract 3: \( 4x = -2\sqrt{6} - 3 \)
Divide by 4: \( x = \frac{-2\sqrt{6} - 3}{4} \)

We can also write this as:

\( x = \frac{-3 \pm 2\sqrt{6}}{4} \)

✔ Final Answer for #3: \( x = \frac{-3 + 2\sqrt{6}}{4} \) or \( x = \frac{-3 - 2\sqrt{6}}{4} \)

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Problem 4: \( (x - 5)^2 + 36 = 0 \)

First, isolate the squared term:

Subtract 36 from both sides:

\( (x - 5)^2 = -36 \)

Now take the square root of both sides:

\( x - 5 = \pm \sqrt{-36} \)

But \( \sqrt{-36} \) is not a real number — it’s imaginary.

\( \sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i \)

So:

\( x - 5 = 6i \) or \( x - 5 = -6i \)

Add 5 to both sides:

\( x = 5 + 6i \) or \( x = 5 - 6i \)

✔ Final Answer for #4: \( x = 5 + 6i \) or \( x = 5 - 6i \)

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Final Answer:
1. \( x = \pm 3\sqrt{3} \)
2. \( x = 5 \) or \( x = -11 \)
3. \( x = \frac{-3 \pm 2\sqrt{6}}{4} \)
4. \( x = 5 \pm 6i \)