Worksheet for solving quadratic equations with four sections and practice problems.

To solve the quadratic equations provided in the image, we will go through each section step by step. Quadratic equations are typically solved using methods such as factoring, completing the square, or the quadratic formula. However, many of these problems can be simplified directly due to their structure.

Section A: Basic Square Root Problems

These problems involve solving equations of the form \( x^2 = k \). The solutions are given by:
\[ x = \pm \sqrt{k} \]

#### Solutions:
1. \( x^2 = 25 \)
\[
x = \pm \sqrt{25} = \pm 5
\]
Answer: \( x = 5, -5 \)

2. \( x^2 = 16 \)
\[
x = \pm \sqrt{16} = \pm 4
\]
Answer: \( x = 4, -4 \)

3. \( x^2 = 81 \)
\[
x = \pm \sqrt{81} = \pm 9
\]
Answer: \( x = 9, -9 \)

4. \( x^2 = 100 \)
\[
x = \pm \sqrt{100} = \pm 10
\]
Answer: \( x = 10, -10 \)

5. \( -x^2 = -64 \)
\[
x^2 = 64 \quad \Rightarrow \quad x = \pm \sqrt{64} = \pm 8
\]
Answer: \( x = 8, -8 \)

6. \( -x^2 = -9 \)
\[
x^2 = 9 \quad \Rightarrow \quad x = \pm \sqrt{9} = \pm 3
\]
Answer: \( x = 3, -3 \)

7. \( -x^2 = -4 \)
\[
x^2 = 4 \quad \Rightarrow \quad x = \pm \sqrt{4} = \pm 2
\]
Answer: \( x = 2, -2 \)

8. \( -x^2 = -49 \)
\[
x^2 = 49 \quad \Rightarrow \quad x = \pm \sqrt{49} = \pm 7
\]
Answer: \( x = 7, -7 \)

9. \( x^2 = 3600 \)
\[
x = \pm \sqrt{3600} = \pm 60
\]
Answer: \( x = 60, -60 \)

10. \( x^2 = 12100 \)
\[
x = \pm \sqrt{12100} = \pm 110
\]
Answer: \( x = 110, -110 \)

11. \( x^2 = 14400 \)
\[
x = \pm \sqrt{14400} = \pm 120
\]
Answer: \( x = 120, -120 \)

12. \( x^2 = 10000 \)
\[
x = \pm \sqrt{10000} = \pm 100
\]
Answer: \( x = 100, -100 \)

---

Section B: Coefficient Multiplication

These problems involve solving equations of the form \( ax^2 = k \). The solutions are given by:
\[ x = \pm \sqrt{\frac{k}{a}} \]

#### Solutions:
1. \( 2x^2 = 2 \)
\[
x^2 = \frac{2}{2} = 1 \quad \Rightarrow \quad x = \pm \sqrt{1} = \pm 1
\]
Answer: \( x = 1, -1 \)

2. \( 2x^2 = 200 \)
\[
x^2 = \frac{200}{2} = 100 \quad \Rightarrow \quad x = \pm \sqrt{100} = \pm 10
\]
Answer: \( x = 10, -10 \)

3. \( 2x^2 = 98 \)
\[
x^2 = \frac{98}{2} = 49 \quad \Rightarrow \quad x = \pm \sqrt{49} = \pm 7
\]
Answer: \( x = 7, -7 \)

4. \( 3x^2 = 27 \)
\[
x^2 = \frac{27}{3} = 9 \quad \Rightarrow \quad x = \pm \sqrt{9} = \pm 3
\]
Answer: \( x = 3, -3 \)

5. \( 3x^2 = 7500 \)
\[
x^2 = \frac{7500}{3} = 2500 \quad \Rightarrow \quad x = \pm \sqrt{2500} = \pm 50
\]
Answer: \( x = 50, -50 \)

6. \( 7x^2 = 2800 \)
\[
x^2 = \frac{2800}{7} = 400 \quad \Rightarrow \quad x = \pm \sqrt{400} = \pm 20
\]
Answer: \( x = 20, -20 \)

7. \( -2x^2 = -162 \)
\[
x^2 = \frac{162}{2} = 81 \quad \Rightarrow \quad x = \pm \sqrt{81} = \pm 9
\]
Answer: \( x = 9, -9 \)

8. \( -4x^2 = -256 \)
\[
x^2 = \frac{256}{4} = 64 \quad \Rightarrow \quad x = \pm \sqrt{64} = \pm 8
\]
Answer: \( x = 8, -8 \)

9. \( (x + 3)^2 = 16 \)
\[
x + 3 = \pm \sqrt{16} = \pm 4
\]
\[
x + 3 = 4 \quad \Rightarrow \quad x = 1
\]
\[
x + 3 = -4 \quad \Rightarrow \quad x = -7
\]
Answer: \( x = 1, -7 \)

10. \( (x + 9)^2 = 36 \)
\[
x + 9 = \pm \sqrt{36} = \pm 6
\]
\[
x + 9 = 6 \quad \Rightarrow \quad x = -3
\]
\[
x + 9 = -6 \quad \Rightarrow \quad x = -15
\]
Answer: \( x = -3, -15 \)

11. \( (x - 4)^2 = 81 \)
\[
x - 4 = \pm \sqrt{81} = \pm 9
\]
\[
x - 4 = 9 \quad \Rightarrow \quad x = 13
\]
\[
x - 4 = -9 \quad \Rightarrow \quad x = -5
\]
Answer: \( x = 13, -5 \)

12. \( -(x - 12)^2 = -121 \)
\[
(x - 12)^2 = 121 \quad \Rightarrow \quad x - 12 = \pm \sqrt{121} = \pm 11
\]
\[
x - 12 = 11 \quad \Rightarrow \quad x = 23
\]
\[
x - 12 = -11 \quad \Rightarrow \quad x = 1
\]
Answer: \( x = 23, 1 \)

---

Section C: General Quadratic Equations

These problems involve solving general quadratic equations. We will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

#### Solutions:
1. \( x^2 + 1 = 2 \)
\[
x^2 = 1 \quad \Rightarrow \quad x = \pm \sqrt{1} = \pm 1
\]
Answer: \( x = 1, -1 \)

2. \( x^2 + 8 = 33 \)
\[
x^2 = 25 \quad \Rightarrow \quad x = \pm \sqrt{25} = \pm 5
\]
Answer: \( x = 5, -5 \)

3. \( x^2 + 27 = 31 \)
\[
x^2 = 4 \quad \Rightarrow \quad x = \pm \sqrt{4} = \pm 2
\]
Answer: \( x = 2, -2 \)

4. \( 31 + x^2 = 152 \)
\[
x^2 = 121 \quad \Rightarrow \quad x = \pm \sqrt{121} = \pm 11
\]
Answer: \( x = 11, -11 \)

5. \( 24 + x^2 = 88 \)
\[
x^2 = 64 \quad \Rightarrow \quad x = \pm \sqrt{64} = \pm 8
\]
Answer: \( x = 8, -8 \)

6. \( 5x^2 + 26 = 106 \)
\[
5x^2 = 80 \quad \Rightarrow \quad x^2 = 16 \quad \Rightarrow \quad x = \pm \sqrt{16} = \pm 4
\]
Answer: \( x = 4, -4 \)

7. \( 15 - x^2 = 14 \)
\[
-x^2 = -1 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm \sqrt{1} = \pm 1
\]
Answer: \( x = 1, -1 \)

8. \( 48 - x^2 = 32 \)
\[
-x^2 = -16 \quad \Rightarrow \quad x^2 = 16 \quad \Rightarrow \quad x = \pm \sqrt{16} = \pm 4
\]
Answer: \( x = 4, -4 \)

9. \( 8 - 2x^2 = -10 \)
\[
-2x^2 = -18 \quad \Rightarrow \quad x^2 = 9 \quad \Rightarrow \quad x = \pm \sqrt{9} = \pm 3
\]
Answer: \( x = 3, -3 \)

10. \( 29 - 3x^2 = -19 \)
\[
-3x^2 = -48 \quad \Rightarrow \quad x^2 = 16 \quad \Rightarrow \quad x = \pm \sqrt{16} = \pm 4
\]
Answer: \( x = 4, -4 \)

11. \( (2x + 7)^2 = 9 \)
\[
2x + 7 = \pm \sqrt{9} = \pm 3
\]
\[
2x + 7 = 3 \quad \Rightarrow \quad 2x = -4 \quad \Rightarrow \quad x = -2
\]
\[
2x + 7 = -3 \quad \Rightarrow \quad 2x = -10 \quad \Rightarrow \quad x = -5
\]
Answer: \( x = -2, -5 \)

12. \( (3x - 5)^2 = 49 \)
\[
3x - 5 = \pm \sqrt{49} = \pm 7
\]
\[
3x - 5 = 7 \quad \Rightarrow \quad 3x = 12 \quad \Rightarrow \quad x = 4
\]
\[
3x - 5 = -7 \quad \Rightarrow \quad 3x = -2 \quad \Rightarrow \quad x = -\frac{2}{3}
\]
Answer: \( x = 4, -\frac{2}{3} \)

13. \( (3x + 7)^2 = 121 \)
\[
3x + 7 = \pm \sqrt{121} = \pm 11
\]
\[
3x + 7 = 11 \quad \Rightarrow \quad 3x = 4 \quad \Rightarrow \quad x = \frac{4}{3}
\]
\[
3x + 7 = -11 \quad \Rightarrow \quad 3x = -18 \quad \Rightarrow \quad x = -6
\]
Answer: \( x = \frac{4}{3}, -6 \)

14. \( (2x - 1)^2 = 4 \)
\[
2x - 1 = \pm \sqrt{4} = \pm 2
\]
\[
2x - 1 = 2 \quad \Rightarrow \quad 2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}
\]
\[
2x - 1 = -2 \quad \Rightarrow \quad 2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}
\]
Answer: \( x = \frac{3}{2}, -\frac{1}{2} \)

15. \( (9 - x)^2 = 25 \)
\[
9 - x = \pm \sqrt{25} = \pm 5
\]
\[
9 - x = 5 \quad \Rightarrow \quad x = 4
\]
\[
9 - x = -5 \quad \Rightarrow \quad x = 14
\]
Answer: \( x = 4, 14 \)

---

Section D: Decimal Precision

These problems involve solving quadratic equations and rounding answers to 2 decimal places.

#### Solutions:
1. \( x^2 = 12 \)
\[
x = \pm \sqrt{12} \approx \pm 3.46
\]
Answer: \( x = 3.46, -3.46 \)

2. \( x^2 = 15 \)
\[
x = \pm \sqrt{15} \approx \pm 3.87
\]
Answer: \( x = 3.87, -3.87 \)

3. \( 6x^2 = 89 \)
\[
x^2 = \frac{89}{6} \approx 14.83 \quad \Rightarrow \quad x = \pm \sqrt{14.83} \approx \pm 3.85
\]
Answer: \( x = 3.85, -3.85 \)

4. \( 9x^2 = 406 \)
\[
x^2 = \frac{406}{9} \approx 45.11 \quad \Rightarrow \quad x = \pm \sqrt{45.11} \approx \pm 6.72
\]
Answer: \( x = 6.72, -6.72 \)

5. \( 7x^2 + 36 = 245 \)
\[
7x^2 = 209 \quad \Rightarrow \quad x^2 = \frac{209}{7} \approx 29.86 \quad \Rightarrow \quad x = \pm \sqrt{29.86} \approx \pm 5.46
\]
Answer: \( x = 5.46, -5.46 \)

6. \( 9x^2 - 41 = 92 \)
\[
9x^2 = 133 \quad \Rightarrow \quad x^2 = \frac{133}{9} \approx 14.78 \quad \Rightarrow \quad x = \pm \sqrt{14.78} \approx \pm 3.84
\]
Answer: \( x = 3.84, -3.84 \)

7. \( 18 - x^2 = 4 \)
\[
-x^2 = -14 \quad \Rightarrow \quad x^2 = 14 \quad \Rightarrow \quad x = \pm \sqrt{14} \approx \pm 3.74
\]
Answer: \( x = 3.74, -3.74 \)

8. \( 144 - 2x^2 = 50 \)
\[
-2x^2 = -94 \quad \Rightarrow \quad x^2 = 47 \quad \Rightarrow \quad x = \pm \sqrt{47} \approx \pm 6.86
\]
Answer: \( x = 6.86, -6.86 \)

9. \( (x - 5)^2 = 32 \)
\[
x - 5 = \pm \sqrt{32} \approx \pm 5.66
\]
\[
x - 5 = 5.66 \quad \Rightarrow \quad x = 10.66
\]
\[
x - 5 = -5.66 \quad \Rightarrow \quad x = -0.66
\]
Answer: \( x = 10.66, -0.66 \)

10. \( (x + 6)^2 = 8 \)
\[
x + 6 = \pm \sqrt{8} \approx \pm 2.83
\]
\[
x + 6 = 2.83 \quad \Rightarrow \quad x = -3.17
\]
\[
x + 6 = -2.83 \quad \Rightarrow \quad x = -8.83
\]
Answer: \( x = -3.17, -8.83 \)

11. \( (x + 1)^2 = 2 \)
\[
x + 1 = \pm \sqrt{2} \approx \pm 1.41
\]
\[
x + 1 = 1.41 \quad \Rightarrow \quad x = 0.41
\]
\[
x + 1 = -1.41 \quad \Rightarrow \quad x = -2.41
\]
Answer: \( x = 0.41, -2.41 \)

12. \( (4x + 11)^2 = 6 \)
\[
4x + 11 = \pm \sqrt{6} \approx \pm 2.45
\]
\[
4x + 11 = 2.45 \quad \Rightarrow \quad 4x = -8.55 \quad \Rightarrow \quad x = -2.14
\]
\[
4x + 11 = -2.45 \quad \Rightarrow \quad 4x = -13.45 \quad \Rightarrow \quad x = -3.36
\]
Answer: \( x = -2.14, -3.36 \)

13. \( (3 + 5x)^2 = 18 \)
\[
3 + 5x = \pm \sqrt{18} \approx \pm 4.24
\]
\[
3 + 5x = 4.24 \quad \Rightarrow \quad 5x = 1.24 \quad \Rightarrow \quad x = 0.25
\]
\[
3 + 5x = -4.24 \quad \Rightarrow \quad 5x = -7.24 \quad \Rightarrow \quad x = -1.45
\]
Answer: \( x = 0.25, -1.45 \)

---

Final Answer:

\[
\boxed{\text{See detailed solutions above.}}
\]