Solving quadratic equations using square roots worksheet with answers.

The worksheet you've uploaded is titled "Solving Quadratic Equations with Square Roots", and it asks students to solve each equation by taking square roots. The equations are all in the form of quadratic equations where the variable is squared, and we can isolate the squared term and then take the square root of both sides.

Let's go through each problem step-by-step and verify the solutions provided (which are already given in red), explaining the process:

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1) $ k^2 = 76 $

- Take the square root of both sides:
$$
k = \pm \sqrt{76}
$$
- Simplify: $ \sqrt{76} \approx 8.717 $
- So: $ k = \pm 8.717 $

✔ Answer: $ \{8.717, -8.717\} $ — Correct

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2) $ k^2 = 16 $

- $ k = \pm \sqrt{16} = \pm 4 $

✔ Answer: $ \{4, -4\} $ — Correct

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3) $ x^2 = 21 $

- $ x = \pm \sqrt{21} \approx \pm 4.582 $

✔ Answer: $ \{4.582, -4.582\} $ — Correct

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4) $ a^2 = 4 $

- $ a = \pm \sqrt{4} = \pm 2 $

✔ Answer: $ \{2, -2\} $ — Correct

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5) $ x^2 + 8 = 28 $

- Subtract 8 from both sides:
$$
x^2 = 20
$$
- $ x = \pm \sqrt{20} \approx \pm 4.472 $

✔ Answer: $ \{4.472, -4.472\} $ — Correct

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6) $ 2n^2 = -144 $

- Divide both sides by 2:
$$
n^2 = -72
$$
- But square of a real number cannot be negative, so no real solution.

✔ Answer: "No solution." — Correct (assuming only real numbers)

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7) $ -6m^2 = -414 $

- Divide both sides by -6:
$$
m^2 = \frac{-414}{-6} = 69
$$
- $ m = \pm \sqrt{69} \approx \pm 8.306 $

✔ Answer: $ \{8.306, -8.306\} $ — Correct

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8) $ 7x^2 = -21 $

- Divide by 7:
$$
x^2 = -3
$$
- Again, no real solution because square root of negative number is not real.

✔ Answer: "No solution." — Correct

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9) $ m^2 + 7 = 88 $

- Subtract 7:
$$
m^2 = 81
$$
- $ m = \pm \sqrt{81} = \pm 9 $

✔ Answer: $ \{9, -9\} $ — Correct

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10) $ -5x^2 = -500 $

- Divide by -5:
$$
x^2 = 100
$$
- $ x = \pm \sqrt{100} = \pm 10 $

✔ Answer: $ \{10, -10\} $ — Correct

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11) $ -7n^2 = -448 $

- Divide by -7:
$$
n^2 = 64
$$
- $ n = \pm \sqrt{64} = \pm 8 $

✔ Answer: $ \{8, -8\} $ — Correct

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12) $ -2k^2 = -162 $

- Divide by -2:
$$
k^2 = 81
$$
- $ k = \pm \sqrt{81} = \pm 9 $

✔ Answer: $ \{9, -9\} $ — Correct

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13) $ x^2 - 5 = 73 $

- Add 5 to both sides:
$$
x^2 = 78
$$
- $ x = \pm \sqrt{78} \approx \pm 8.831 $

✔ Answer: $ \{8.831, -8.831\} $ — Correct

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14) $ 16n^2 = 49 $

- Divide by 16:
$$
n^2 = \frac{49}{16}
$$
- $ n = \pm \sqrt{\frac{49}{16}} = \pm \frac{7}{4} = \pm 1.75 $

✔ Answer: $ \{1.75, -1.75\} $ — Correct

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✔ Summary:

All answers on the worksheet are correct!

🔍 Key Concepts Used:

- To solve equations like $ ax^2 = c $, isolate $ x^2 $ first.
- Then take the square root of both sides, remembering that:
$$
x = \pm \sqrt{\text{value}}
$$
- If the value under the square root is negative, there is no real solution.
- Always check if the answer makes sense in context (especially for real vs complex numbers).

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