Quadratic equations worksheet with square roots, where students solve equations and color by number based on the positive root.

Let's solve each quadratic equation by isolating \( x^2 \) and then taking the square root. We are only interested in the positive root for coloring, as per the directions.

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1. \( x^2 + 6 = 6 \)

Subtract 6 from both sides:

\( x^2 = 0 \)

Take square root:

\( x = \sqrt{0} = 0 \)

✔ Positive root: 0 — Color: Yellow

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2. \( x^2 + 4 = 40 \)

Subtract 4:

\( x^2 = 36 \)

Take square root:

\( x = \sqrt{36} = 6 \)

✔ Positive root: 6 — Color: Purple

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3. \( -3x^2 = 147 \)

Divide both sides by -3:

\( x^2 = -49 \)

Take square root:

\( x = \sqrt{-49} \) → Not real (imaginary)

But the worksheet says “solve by taking the square root” and “leave answers in simplest fraction/radical form.” Since it’s imaginary, and we’re coloring a picture (likely expecting real numbers), this might be an error or trick. However, if we proceed formally:

\( x = \pm \sqrt{-49} = \pm 7i \)

There is no positive real root. But since the instructions say to color the *positive* root, and there isn’t one, perhaps skip? Or maybe the problem expects us to note no real solution.

However, let’s check the original problem again — maybe I misread.

Wait — -3x² = 147 → x² = -49 → no real solution.

So technically, no positive real root. But since the worksheet likely expects real roots, perhaps double-check if the equation was meant to be different? But as written, we must report:

✘ No real positive root — but the instruction says to color the positive root. This may be a trick question. Let’s leave it blank for now and come back if needed.

But wait — perhaps the problem is designed to have real solutions only? Let me verify others first.

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4. \( -5x^2 + 2 = -1088 \)

Subtract 2:

\( -5x^2 = -1090 \)

Divide by -5:

\( x^2 = 218 \)

Take square root:

\( x = \sqrt{218} \)

Can we simplify? 218 = 2 × 109 → both prime → cannot simplify.

✔ Positive root: \( \sqrt{218} \) — Color: Blue

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5. \( x^2 + 2 = 16 \)

Subtract 2:

\( x^2 = 14 \)

Take square root:

\( x = \sqrt{14} \)

✔ Positive root: \( \sqrt{14} \) — Color: Black

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6. \( -10 - 5x^2 = -330 \)

Add 10 to both sides:

\( -5x^2 = -320 \)

Divide by -5:

\( x^2 = 64 \)

Take square root:

\( x = \sqrt{64} = 8 \)

✔ Positive root: 8 — Color: Yellow

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7. \( -8x^2 - 7 = 193 \)

Add 7:

\( -8x^2 = 200 \)

Divide by -8:

\( x^2 = -25 \)

Again, negative → imaginary.

\( x = \pm 5i \)

✘ No real positive root.

Same issue as #3. Perhaps these are mistakes? Or maybe we're supposed to write the principal square root even if complex? But coloring suggests real numbers.

We’ll flag these and continue.

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8. \( 8x^2 + 4 = 164 \)

Subtract 4:

\( 8x^2 = 160 \)

Divide by 8:

\( x^2 = 20 \)

Take square root:

\( x = \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \)

✔ Positive root: \( 2\sqrt{5} \) — Color: Green

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9. \( 6x^2 + 7 = 607 \)

Subtract 7:

\( 6x^2 = 600 \)

Divide by 6:

\( x^2 = 100 \)

Take square root:

\( x = \sqrt{100} = 10 \)

✔ Positive root: 10 — Color: Blue

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10. \( -9x^2 + 10 = 91 \)

Subtract 10:

\( -9x^2 = 81 \)

Divide by -9:

\( x^2 = -9 \)

Again, negative → imaginary.

\( x = \pm 3i \)

✘ No real positive root.

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11. \( 13 - 8x^2 = -1139 \)

Subtract 13:

\( -8x^2 = -1152 \)

Divide by -8:

\( x^2 = 144 \)

Take square root:

\( x = \sqrt{144} = 12 \)

✔ Positive root: 12 — Color: Yellow

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12. \( -4x^2 = 96 \)

Divide by -4:

\( x^2 = -24 \)

Again, negative → imaginary.

\( x = \pm \sqrt{-24} = \pm 2i\sqrt{6} \)

✘ No real positive root.

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## Summary of Positive Real Roots:

| Problem | Positive Root | Color |
|---------|----------------------|---------|
| 1 | 0 | Yellow |
| 2 | 6 | Purple |
| 3 | ✘ No real root | Green |
| 4 | √218 | Blue |
| 5 | √14 | Black |
| 6 | 8 | Yellow |
| 7 | ✘ No real root | Purple |
| 8 | 2√5 | Green |
| 9 | 10 | Blue |
| 10 | ✘ No real root | Black |
| 11 | 12 | Yellow |
| 12 | ✘ No real root | Blue |

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## Final Answer (for coloring):

You should fill in the "Square Roots" column with the positive real roots where they exist:

1. 0
2. 6
3. *(No real root — perhaps leave blank or write “no real solution”)*
4. √218
5. √14
6. 8
7. *(No real root)*
8. 2√5
9. 10
10. *(No real root)*
11. 12
12. *(No real root)*

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## Coloring Instructions:

Color the positive root on the picture using the indicated color:

- Yellow: Problems 1, 6, 11 → roots: 0, 8, 12
- Purple: Problems 2, 7 → root: 6 (and 7 has no real root — maybe skip or color nothing)
- Green: Problems 3, 8 → root: 2√5 (and 3 has no real root)
- Blue: Problems 4, 9, 12 → roots: √218, 10 (and 12 has no real root)
- Black: Problems 5, 10 → root: √14 (and 10 has no real root)

⚠️ Note: Problems 3, 7, 10, and 12 have no real positive roots, so you may need to leave those uncolored or consult your teacher. It’s possible there’s a typo in the worksheet (e.g., sign errors). For example, if problem 3 were `3x² = 147`, then x = 7 — which would make sense.

But based on what’s written, we solved correctly.

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✔ Final Boxed Answers for Square Roots Column (only real positive roots):

1. \( \boxed{0} \)
2. \( \boxed{6} \)
3. No real solution
4. \( \boxed{\sqrt{218}} \)
5. \( \boxed{\sqrt{14}} \)
6. \( \boxed{8} \)
7. No real solution
8. \( \boxed{2\sqrt{5}} \)
9. \( \boxed{10} \)
10. No real solution
11. \( \boxed{12} \)
12. No real solution

Use these to color the picture accordingly.