Edia | Free math homework in minutes - Free Printable

Let’s solve each problem one by one using the square root property. That means we’ll isolate the squared term, then take the square root of both sides — remembering to include both the positive and negative roots!

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Problem 1: Solve for $ m $.

$ 4m^2 + 4 = 34 $

Step 1: Subtract 4 from both sides.
→ $ 4m^2 = 30 $

Step 2: Divide both sides by 4.
→ $ m^2 = \frac{30}{4} = \frac{15}{2} $

Step 3: Take square root of both sides.
→ $ m = \pm \sqrt{\frac{15}{2}} $

Step 4: Simplify radical (rationalize denominator).
→ $ m = \pm \frac{\sqrt{30}}{2} $

✔ Final Answer for #1: $ \boxed{m = \pm \frac{\sqrt{30}}{2}} $

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Problem 2: Solve for $ k $.

$ 36k^2 - 10 = 7 $

Step 1: Add 10 to both sides.
→ $ 36k^2 = 17 $

Step 2: Divide by 36.
→ $ k^2 = \frac{17}{36} $

Step 3: Square root both sides.
→ $ k = \pm \sqrt{\frac{17}{36}} = \pm \frac{\sqrt{17}}{6} $

✔ Final Answer for #2: $ \boxed{k = \pm \frac{\sqrt{17}}{6}} $

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Problem 3: Solve for $ m $.

$ 25m^2 - 10 = 13 $

Step 1: Add 10 to both sides.
→ $ 25m^2 = 23 $

Step 2: Divide by 25.
→ $ m^2 = \frac{23}{25} $

Step 3: Square root.
→ $ m = \pm \frac{\sqrt{23}}{5} $

✔ Final Answer for #3: $ \boxed{m = \pm \frac{\sqrt{23}}{5}} $

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Problem 4: Solve for $ k $.

$ 16k^2 - 11 = 34 $

Step 1: Add 11 to both sides.
→ $ 16k^2 = 45 $

Step 2: Divide by 16.
→ $ k^2 = \frac{45}{16} $

Step 3: Square root.
→ $ k = \pm \sqrt{\frac{45}{16}} = \pm \frac{\sqrt{45}}{4} $

Step 4: Simplify $ \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} $
→ $ k = \pm \frac{3\sqrt{5}}{4} $

✔ Final Answer for #4: $ \boxed{k = \pm \frac{3\sqrt{5}}{4}} $

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Problem 5: Solve for $ n $.

$ \left(n - \frac{1}{2}\right)^2 = 10 $

Step 1: Take square root of both sides.
→ $ n - \frac{1}{2} = \pm \sqrt{10} $

Step 2: Add $ \frac{1}{2} $ to both sides.
→ $ n = \frac{1}{2} \pm \sqrt{10} $

We can write as a single fraction if needed, but this is simplified.

✔ Final Answer for #5: $ \boxed{n = \frac{1}{2} \pm \sqrt{10}} $

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Problem 6: Solve for $ z $.

$ \left(z - \frac{4}{3}\right)^2 = 9 $

Step 1: Take square root.
→ $ z - \frac{4}{3} = \pm 3 $

Step 2: Add $ \frac{4}{3} $ to both sides.

Case 1: $ z = \frac{4}{3} + 3 = \frac{4}{3} + \frac{9}{3} = \frac{13}{3} $

Case 2: $ z = \frac{4}{3} - 3 = \frac{4}{3} - \frac{9}{3} = -\frac{5}{3} $

✔ Final Answer for #6: $ \boxed{z = \frac{13}{3},\ -\frac{5}{3}} $

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Problem 7: Solve for $ n $.

$ \left(n + \frac{1}{4}\right)^2 = 2 $

Step 1: Take square root.
→ $ n + \frac{1}{4} = \pm \sqrt{2} $

Step 2: Subtract $ \frac{1}{4} $.
→ $ n = -\frac{1}{4} \pm \sqrt{2} $

✔ Final Answer for #7: $ \boxed{n = -\frac{1}{4} \pm \sqrt{2}} $

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Problem 8: Solve for $ m $. Write in terms of $ i $ if complex.

$ m^2 - 4m + 4 = -2 $

Notice: Left side is a perfect square trinomial!

→ $ (m - 2)^2 = -2 $

Step 1: Take square root.
→ $ m - 2 = \pm \sqrt{-2} = \pm i\sqrt{2} $

Step 2: Add 2 to both sides.
→ $ m = 2 \pm i\sqrt{2} $

✔ Final Answer for #8: $ \boxed{m = 2 \pm i\sqrt{2}} $

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Final Answer:
1. $ m = \pm \frac{\sqrt{30}}{2} $
2. $ k = \pm \frac{\sqrt{17}}{6} $
3. $ m = \pm \frac{\sqrt{23}}{5} $
4. $ k = \pm \frac{3\sqrt{5}}{4} $
5. $ n = \frac{1}{2} \pm \sqrt{10} $
6. $ z = \frac{13}{3},\ -\frac{5}{3} $
7. $ n = -\frac{1}{4} \pm \sqrt{2} $
8. $ m = 2 \pm i\sqrt{2} $