Math worksheet for solving quadratic equations using the square root property.

Let’s solve each quadratic equation using the Square Root Property.

The Square Root Property says:
If \( x^2 = k \), then \( x = \sqrt{k} \) or \( x = -\sqrt{k} \).
We can write this as \( x = \pm \sqrt{k} \).

We’ll go one problem at a time.

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1) \( n^2 - 12 = 0 \)

Add 12 to both sides:
\( n^2 = 12 \)

Take square root of both sides:
\( n = \pm \sqrt{12} \)

Simplify \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \)

✔ Final Answer for #1: \( n = \pm 2\sqrt{3} \)

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2) \( p^2 = -36 \)

Take square root:
\( p = \pm \sqrt{-36} \)

But square root of negative number is not real — it’s imaginary.
\( \sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i \)

✔ Final Answer for #2: \( p = \pm 6i \)

*(Note: If your class hasn’t learned imaginary numbers yet, you might say “no real solution.” But since the problem doesn’t specify, we’ll include complex solutions.)*

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3) \( (x - 6)^2 = 49 \)

Take square root of both sides:
\( x - 6 = \pm \sqrt{49} = \pm 7 \)

Now solve for x:

Case 1: \( x - 6 = 7 \) → \( x = 13 \)
Case 2: \( x - 6 = -7 \) → \( x = -1 \)

✔ Final Answer for #3: \( x = 13 \) or \( x = -1 \)

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4) \( r^2 - 32 = 0 \)

Add 32:
\( r^2 = 32 \)

Take square root:
\( r = \pm \sqrt{32} \)

Simplify: \( \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \)

✔ Final Answer for #4: \( r = \pm 4\sqrt{2} \)

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5) \( u^2 = 64 \)

Take square root:
\( u = \pm \sqrt{64} = \pm 8 \)

✔ Final Answer for #5: \( u = \pm 8 \)

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6) \( (b + 1)^2 = 16 \)

Take square root:
\( b + 1 = \pm \sqrt{16} = \pm 4 \)

Solve:

Case 1: \( b + 1 = 4 \) → \( b = 3 \)
Case 2: \( b + 1 = -4 \) → \( b = -5 \)

✔ Final Answer for #6: \( b = 3 \) or \( b = -5 \)

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7) \( (y + 2)^2 = -9 \)

Take square root:
\( y + 2 = \pm \sqrt{-9} = \pm 3i \)

Subtract 2:
\( y = -2 \pm 3i \)

✔ Final Answer for #7: \( y = -2 + 3i \) or \( y = -2 - 3i \)

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8) \( m^2 + 63 = 0 \)

Subtract 63:
\( m^2 = -63 \)

Take square root:
\( m = \pm \sqrt{-63} = \pm \sqrt{63} \cdot i \)

Simplify \( \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \)

So: \( m = \pm 3\sqrt{7}\,i \)

✔ Final Answer for #8: \( m = \pm 3\sqrt{7}\,i \)

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9) \( w^2 + 72 = 0 \)

Subtract 72:
\( w^2 = -72 \)

Take square root:
\( w = \pm \sqrt{-72} = \pm \sqrt{72} \cdot i \)

Simplify \( \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \)

So: \( w = \pm 6\sqrt{2}\,i \)

✔ Final Answer for #9: \( w = \pm 6\sqrt{2}\,i \)

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10) \( q^2 - 25 = 0 \)

Add 25:
\( q^2 = 25 \)

Take square root:
\( q = \pm \sqrt{25} = \pm 5 \)

✔ Final Answer for #10: \( q = \pm 5 \)

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Final Answer:
1) \( n = \pm 2\sqrt{3} \)
2) \( p = \pm 6i \)
3) \( x = 13 \) or \( x = -1 \)
4) \( r = \pm 4\sqrt{2} \)
5) \( u = \pm 8 \)
6) \( b = 3 \) or \( b = -5 \)
7) \( y = -2 \pm 3i \)
8) \( m = \pm 3\sqrt{7}\,i \)
9) \( w = \pm 6\sqrt{2}\,i \)
10) \( q = \pm 5 \)