SOLUTION: Solving Polynomial Equations Analysis Problem Set ... - Free Printable

Actually, there is a critical error in the student’s work on Problem 1, and several other problems also contain mistakes or incomplete reasoning. Let’s go through each problem carefully, correct the errors, and explain the proper solutions.

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## ✔ General Reminder:
Before factoring, make sure the equation is set equal to zero:
> `ax² + bx + c = 0`

Then factor, and use the Zero Product Property:
> If `(x - a)(x - b) = 0`, then `x = a` or `x = b`.

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# 🔍 Problem-by-Problem Corrections & Explanations

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1.) x² + 5x + 6 = 0

✔ Student’s work:
`(x - 2)(x - 3) = 0` → `x = 2, 3` ✘ WRONG FACTORING

✘ Mistake:
The student wrote `(x - 2)(x - 3)` which expands to `x² - 5x + 6`, but we need `+5x`, not `-5x`.

✔ Correct Factoring:
We need two numbers that multiply to +6 and add to +5 → 2 and 3

So:
`(x + 2)(x + 3) = 0`

→ `x + 2 = 0` → `x = -2`
→ `x + 3 = 0` → `x = -3`

✔ Correct Answer: `x = -2, -3`

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2.) x² - 10x = 0

✔ Student’s work:
`x(x - 10) = 0` → `x = 0, 10` ✔ CORRECT

This is correct — factored out GCF (x), then used zero product property.

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3.) x² - 21x = -80

✔ Student moved -80 to left:
`x² - 21x + 80 = 0` ✔ Correct setup

✔ Factored as `(x - 5)(x - 16) = 0` → `x = 5, 16` ✔ CORRECT

Check: `(-5)*(-16) = 80`, `-5 + (-16) = -21` → Perfect.

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4.) x² = 8x - 16

✔ Student rearranged:
`x² - 8x + 16 = 0` ✔ Correct

✔ Factored: `(x - 4)² = 0` → `x = 4` ✔ CORRECT

This is a perfect square trinomial. Only one solution (double root).

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5.) x² - 25 = 0

✔ Student wrote: `x² = 25` → `x = ±√25` → `x = 5, -5` ✔ CORRECT

This is a difference of squares: `(x - 5)(x + 5) = 0`

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6.) 49x² - 36 = 0

✔ Student solved:
`49x² = 36` → `x² = 36/49` → `x = ±6/7` ✔ CORRECT

Difference of squares: `(7x - 6)(7x + 6) = 0`

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7.) 9x² = 36

✔ Student divided both sides by 9:
`x² = 4` → `x = ±2` → `x = 2, -2` ✔ CORRECT

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8.) 3x² + 15x = 42

✔ Student moved 42 to left:
`3x² + 15x - 42 = 0` ✔ Correct

✔ Factored out GCF 3:
`3(x² + 5x - 14) = 0`

Now factor inside: Need two numbers that multiply to -14, add to +5 → 7 and -2

So: `3(x + 7)(x - 2) = 0`

→ `x + 7 = 0` → `x = -7`
→ `x - 2 = 0` → `x = 2`

✔ Answer: `x = 2, -7` — Student got this right! ✔

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9.) Free space — No problem assigned.

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## 📌 Problems 10–12 (Bottom Row)

10.) x² + 10x + 21 = 0

✔ Student factored: `(x + 3)(x + 7) = 0` → `x = -3, -7` ✔ CORRECT

Check: 3*7=21, 3+7=10 → Good.

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11.) 6x² + 5x - 4 = 0

✔ Student factored: `(2x - 1)(3x + 4) = 0`

Let’s check:
`(2x)(3x) = 6x²`
`(2x)(4) + (-1)(3x) = 8x - 3x = 5x`
`(-1)(4) = -4` → Perfect!

Solutions:
`2x - 1 = 0` → `x = 1/2`
`3x + 4 = 0` → `x = -4/3`

✔ Answer: `x = 1/2, -4/3` — Student wrote `x = 1/2, -4/3` — CORRECT!

*(Note: Student wrote “x = ½, -⁴⁄₃” — same thing.)*

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12.) 0 = x² - 3x

✔ Student wrote: `x(x - 3) = 0` → `x = 0, 3` ✔ CORRECT

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## 📐 Problems 13–18 (Square Roots Section)

> *“Solve each equation using square roots.”*

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13.) x² = -144

✔ Student: `x = √(-144)` → `x = ±12i` ✔ CORRECT

Imaginary numbers: `√(-144) = √(144) * √(-1) = 12i`

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14.) 2x² = 400

✔ Student: `x² = 200` → `x = ±√200 = ±10√2` ✔ CORRECT

`√200 = √(100 * 2) = 10√2`

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15.) x² - 100 = -82

✔ Student: `x² = 18` → `x = ±√18 = ±3√2` ✔ CORRECT

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16.) ½(x - 4)² = 3

✔ Student multiplied both sides by 2:
`(x - 4)² = 6` → Wait, no — they wrote `= 9` ✘ MISTAKE

Let’s fix:

`½(x - 4)² = 3`
Multiply both sides by 2:
`(x - 4)² = 6`

Then:
`x - 4 = ±√6`
→ `x = 4 ± √6`

✔ Correct Answer: `x = 4 + √6, 4 - √6`

Student incorrectly said `(x - 4)² = 9` → leading to `x = 7, 1`. That’s wrong.

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17.) 4(x + 1)² = -100

✔ Student: `(x + 1)² = -25` → `x + 1 = ±√(-25)` → `x = -1 ± 5i`

Wait — student wrote `x = 5i - 1, -5i - 1` → which is same as `-1 ± 5i` ✔ CORRECT

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18.) ¼(x - 8)² = 7

✔ Student: `(x - 8)² = 28` → `x - 8 = ±√28 = ±2√7` → `x = 8 ± 2√7`

Student wrote: `x = 2√7 + 8, -2√7 + 8` → same thing ✔ CORRECT

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## 🧾 Summary of Errors:

| Problem | Error |
|--------|-------|
| 1 | Factored as `(x-2)(x-3)` instead of `(x+2)(x+3)` → Wrong signs |
| 16 | Miscomputed `½(x-4)² = 3` → thought it became `(x-4)² = 9` → should be `=6` |

All other answers are correct.

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## ✔ Final Correct Answers:

1. x = -2, -3
2. x = 0, 10
3. x = 5, 16
4. x = 4
5. x = 5, -5
6. x = 6/7, -6/7
7. x = 2, -2
8. x = 2, -7
9. —
10. x = -3, -7
11. x = 1/2, -4/3
12. x = 0, 3
13. x = 12i, -12i
14. x = 10√2, -10√2
15. x = 3√2, -3√2
16. x = 4 + √6, 4 - √6 *(student error)*
17. x = -1 + 5i, -1 - 5i
18. x = 8 + 2√7, 8 - 2√7

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## 💡 Tip for Student:
Always check your factoring by multiplying back out!
For example, if you think `(x-2)(x-3) = x² +5x+6`, multiply it:
→ `x² -5x +6` — that’s NOT what you started with. So you know it’s wrong.

Keep up the good work — most of your math is spot-on! Just watch those signs and double-check arithmetic steps. 😊