Worksheet on completing the square with equations and solutions.

Problem Overview:

The task involves solving quadratic equations using different methods, including completing the square and applying the quadratic formula. Let's solve each section step by step.

---

Section A: Solve the following equations.

#### 1) \((x + 7)^2 - 9 = 0\)
\[
(x + 7)^2 - 9 = 0
\]
Add 9 to both sides:
\[
(x + 7)^2 = 9
\]
Take the square root of both sides:
\[
x + 7 = \pm 3
\]
Solve for \(x\):
\[
x + 7 = 3 \quad \text{or} \quad x + 7 = -3
\]
\[
x = 3 - 7 \quad \text{or} \quad x = -3 - 7
\]
\[
x = -4 \quad \text{or} \quad x = -10
\]

Solution: \(\boxed{-4, -10}\)

---

#### 2) \((x - 4)^2 - 25 = 0\)
\[
(x - 4)^2 - 25 = 0
\]
Add 25 to both sides:
\[
(x - 4)^2 = 25
\]
Take the square root of both sides:
\[
x - 4 = \pm 5
\]
Solve for \(x\):
\[
x - 4 = 5 \quad \text{or} \quad x - 4 = -5
\]
\[
x = 5 + 4 \quad \text{or} \quad x = -5 + 4
\]
\[
x = 9 \quad \text{or} \quad x = -1
\]

Solution: \(\boxed{9, -1}\)

---

#### 3) \(2(x + 5)^2 - 32 = 0\)
\[
2(x + 5)^2 - 32 = 0
\]
Add 32 to both sides:
\[
2(x + 5)^2 = 32
\]
Divide by 2:
\[
(x + 5)^2 = 16
\]
Take the square root of both sides:
\[
x + 5 = \pm 4
\]
Solve for \(x\):
\[
x + 5 = 4 \quad \text{or} \quad x + 5 = -4
\]
\[
x = 4 - 5 \quad \text{or} \quad x = -4 - 5
\]
\[
x = -1 \quad \text{or} \quad x = -9
\]

Solution: \(\boxed{-1, -9}\)

---

#### 4) \(5(x + 4)^2 - 180 = 0\)
\[
5(x + 4)^2 - 180 = 0
\]
Add 180 to both sides:
\[
5(x + 4)^2 = 180
\]
Divide by 5:
\[
(x + 4)^2 = 36
\]
Take the square root of both sides:
\[
x + 4 = \pm 6
\]
Solve for \(x\):
\[
x + 4 = 6 \quad \text{or} \quad x + 4 = -6
\]
\[
x = 6 - 4 \quad \text{or} \quad x = -6 - 4
\]
\[
x = 2 \quad \text{or} \quad x = -10
\]

Solution: \(\boxed{2, -10}\)

---

Section B: Solve the following equations by completing the square. Leave your answers in surd form.

#### 1) \(x^2 + 4x - 30 = 0\)
Rewrite the equation:
\[
x^2 + 4x = 30
\]
Complete the square:
\[
x^2 + 4x + 4 = 30 + 4
\]
\[
(x + 2)^2 = 34
\]
Take the square root of both sides:
\[
x + 2 = \pm \sqrt{34}
\]
Solve for \(x\):
\[
x = -2 \pm \sqrt{34}
\]

Solution: \(\boxed{-2 \pm \sqrt{34}}\)

---

#### 2) \(x^2 + 18x - 21 = 0\)
Rewrite the equation:
\[
x^2 + 18x = 21
\]
Complete the square:
\[
x^2 + 18x + 81 = 21 + 81
\]
\[
(x + 9)^2 = 102
\]
Take the square root of both sides:
\[
x + 9 = \pm \sqrt{102}
\]
Solve for \(x\):
\[
x = -9 \pm \sqrt{102}
\]

Solution: \(\boxed{-9 \pm \sqrt{102}}\)

---

#### 3) \(x^2 - 6x - 3 = 0\)
Rewrite the equation:
\[
x^2 - 6x = 3
\]
Complete the square:
\[
x^2 - 6x + 9 = 3 + 9
\]
\[
(x - 3)^2 = 12
\]
Take the square root of both sides:
\[
x - 3 = \pm \sqrt{12}
\]
Simplify \(\sqrt{12}\):
\[
\sqrt{12} = 2\sqrt{3}
\]
Solve for \(x\):
\[
x = 3 \pm 2\sqrt{3}
\]

Solution: \(\boxed{3 \pm 2\sqrt{3}}\)

---

#### 4) \(x^2 - x - 7 = 0\)
Rewrite the equation:
\[
x^2 - x = 7
\]
Complete the square:
\[
x^2 - x + \left(\frac{1}{2}\right)^2 = 7 + \left(\frac{1}{2}\right)^2
\]
\[
x^2 - x + \frac{1}{4} = 7 + \frac{1}{4}
\]
\[
\left(x - \frac{1}{2}\right)^2 = \frac{28}{4} + \frac{1}{4}
\]
\[
\left(x - \frac{1}{2}\right)^2 = \frac{29}{4}
\]
Take the square root of both sides:
\[
x - \frac{1}{2} = \pm \sqrt{\frac{29}{4}}
\]
Simplify:
\[
x - \frac{1}{2} = \pm \frac{\sqrt{29}}{2}
\]
Solve for \(x\):
\[
x = \frac{1}{2} \pm \frac{\sqrt{29}}{2}
\]
Combine terms:
\[
x = \frac{1 \pm \sqrt{29}}{2}
\]

Solution: \(\boxed{\frac{1 \pm \sqrt{29}}{2}}\)

---

#### 5) \(x^2 + 12x - 5 = 23\)
Rewrite the equation:
\[
x^2 + 12x - 5 - 23 = 0
\]
\[
x^2 + 12x - 28 = 0
\]
Rewrite:
\[
x^2 + 12x = 28
\]
Complete the square:
\[
x^2 + 12x + 36 = 28 + 36
\]
\[
(x + 6)^2 = 64
\]
Take the square root of both sides:
\[
x + 6 = \pm 8
\]
Solve for \(x\):
\[
x = -6 \pm 8
\]
\[
x = 2 \quad \text{or} \quad x = -14
\]

Solution: \(\boxed{2, -14}\)

---

#### 6) \(x^2 - 32x + 45 = -35\)
Rewrite the equation:
\[
x^2 - 32x + 45 + 35 = 0
\]
\[
x^2 - 32x + 80 = 0
\]
Rewrite:
\[
x^2 - 32x = -80
\]
Complete the square:
\[
x^2 - 32x + 256 = -80 + 256
\]
\[
(x - 16)^2 = 176
\]
Take the square root of both sides:
\[
x - 16 = \pm \sqrt{176}
\]
Simplify \(\sqrt{176}\):
\[
\sqrt{176} = \sqrt{16 \cdot 11} = 4\sqrt{11}
\]
Solve for \(x\):
\[
x = 16 \pm 4\sqrt{11}
\]

Solution: \(\boxed{16 \pm 4\sqrt{11}}\)

---

#### 7) \(x^2 + 3x - 24 = 1\)
Rewrite the equation:
\[
x^2 + 3x - 24 - 1 = 0
\]
\[
x^2 + 3x - 25 = 0
\]
Rewrite:
\[
x^2 + 3x = 25
\]
Complete the square:
\[
x^2 + 3x + \left(\frac{3}{2}\right)^2 = 25 + \left(\frac{3}{2}\right)^2
\]
\[
x^2 + 3x + \frac{9}{4} = 25 + \frac{9}{4}
\]
\[
\left(x + \frac{3}{2}\right)^2 = \frac{100}{4} + \frac{9}{4}
\]
\[
\left(x + \frac{3}{2}\right)^2 = \frac{109}{4}
\]
Take the square root of both sides:
\[
x + \frac{3}{2} = \pm \sqrt{\frac{109}{4}}
\]
Simplify:
\[
x + \frac{3}{2} = \pm \frac{\sqrt{109}}{2}
\]
Solve for \(x\):
\[
x = -\frac{3}{2} \pm \frac{\sqrt{109}}{2}
\]
Combine terms:
\[
x = \frac{-3 \pm \sqrt{109}}{2}
\]

Solution: \(\boxed{\frac{-3 \pm \sqrt{109}}{2}}\)

---

#### 8) \(x^2 - 5x - 4 = 5\)
Rewrite the equation:
\[
x^2 - 5x - 4 - 5 = 0
\]
\[
x^2 - 5x - 9 = 0
\]
Rewrite:
\[
x^2 - 5x = 9
\]
Complete the square:
\[
x^2 - 5x + \left(\frac{5}{2}\right)^2 = 9 + \left(\frac{5}{2}\right)^2
\]
\[
x^2 - 5x + \frac{25}{4} = 9 + \frac{25}{4}
\]
\[
\left(x - \frac{5}{2}\right)^2 = \frac{36}{4} + \frac{25}{4}
\]
\[
\left(x - \frac{5}{2}\right)^2 = \frac{61}{4}
\]
Take the square root of both sides:
\[
x - \frac{5}{2} = \pm \sqrt{\frac{61}{4}}
\]
Simplify:
\[
x - \frac{5}{2} = \pm \frac{\sqrt{61}}{2}
\]
Solve for \(x\):
\[
x = \frac{5}{2} \pm \frac{\sqrt{61}}{2}
\]
Combine terms:
\[
x = \frac{5 \pm \sqrt{61}}{2}
\]

Solution: \(\boxed{\frac{5 \pm \sqrt{61}}{2}}\)

---

Section C: Solve the following equations by completing the square. Leave your answers to one decimal place.

#### 1) \(2x^2 + 4x - 18 = 0\)
Divide the entire equation by 2:
\[
x^2 + 2x - 9 = 0
\]
Rewrite:
\[
x^2 + 2x = 9
\]
Complete the square:
\[
x^2 + 2x + 1 = 9 + 1
\]
\[
(x + 1)^2 = 10
\]
Take the square root of both sides:
\[
x + 1 = \pm \sqrt{10}
\]
Solve for \(x\):
\[
x = -1 \pm \sqrt{10}
\]
Approximate \(\sqrt{10} \approx 3.162\):
\[
x = -1 + 3.162 \quad \text{or} \quad x = -1 - 3.162
\]
\[
x \approx 2.2 \quad \text{or} \quad x \approx -4.2
\]

Solution: \(\boxed{2.2, -4.2}\)

---

#### 2) \(2x^2 + 8x - 5 = 0\)
Divide the entire equation by 2:
\[
x^2 + 4x - \frac{5}{2} = 0
\]
Rewrite:
\[
x^2 + 4x = \frac{5}{2}
\]
Complete the square:
\[
x^2 + 4x + 4 = \frac{5}{2} + 4
\]
\[
(x + 2)^2 = \frac{5}{2} + \frac{8}{2}
\]
\[
(x + 2)^2 = \frac{13}{2}
\]
Take the square root of both sides:
\[
x + 2 = \pm \sqrt{\frac{13}{2}}
\]
Simplify:
\[
x + 2 = \pm \frac{\sqrt{26}}{2}
\]
Solve for \(x\):
\[
x = -2 \pm \frac{\sqrt{26}}{2}
\]
Approximate \(\sqrt{26} \approx 5.10\):
\[
x = -2 + \frac{5.10}{2} \quad \text{or} \quad x = -2 - \frac{5.10}{2}
\]
\[
x \approx -2 + 2.55 \quad \text{or} \quad x \approx -2 - 2.55
\]
\[
x \approx 0.6 \quad \text{or} \quad x \approx -4.6
\]

Solution: \(\boxed{0.6, -4.6}\)

---

#### 3) \(3x^2 - 12x + 4 = 0\)
Divide the entire equation by 3:
\[
x^2 - 4x + \frac{4}{3} = 0
\]
Rewrite:
\[
x^2 - 4x = -\frac{4}{3}
\]
Complete the square:
\[
x^2 - 4x + 4 = -\frac{4}{3} + 4
\]
\[
(x - 2)^2 = -\frac{4}{3} + \frac{12}{3}
\]
\[
(x - 2)^2 = \frac{8}{3}
\]
Take the square root of both sides:
\[
x - 2 = \pm \sqrt{\frac{8}{3}}
\]
Simplify:
\[
x - 2 = \pm \frac{\sqrt{24}}{3}
\]
\[
x - 2 = \pm \frac{2\sqrt{6}}{3}
\]
Solve for \(x\):
\[
x = 2 \pm \frac{2\sqrt{6}}{3}
\]
Approximate \(\sqrt{6} \approx 2.45\):
\[
x = 2 + \frac{2 \cdot 2.45}{3} \quad \text{or} \quad x = 2 - \frac{2 \cdot 2.45}{3}
\]
\[
x \approx 2 + \frac{4.90}{3} \quad \text{or} \quad x \approx 2 - \frac{4.90}{3}
\]
\[
x \approx 2 + 1.63 \quad \text{or} \quad x \approx 2 - 1.63
\]
\[
x \approx 3.6 \quad \text{or} \quad x \approx 0.4
\]

Solution: \(\boxed{3.6, 0.4}\)

---

#### 4) \(2x^2 + 5x - 10 = 0\)
Divide the entire equation by 2:
\[
x^2 + \frac{5}{2}x - 5 = 0
\]
Rewrite:
\[
x^2 + \frac{5}{2}x = 5
\]
Complete the square:
\[
x^2 + \frac{5}{2}x + \left(\frac{5}{4}\right)^2 = 5 + \left(\frac{5}{4}\right)^2
\]
\[
x^2 + \frac{5}{2}x + \frac{25}{16} = 5 + \frac{25}{16}
\]
\[
\left(x + \frac{5}{4}\right)^2 = \frac{80}{16} + \frac{25}{16}
\]
\[
\left(x + \frac{5}{4}\right)^2 = \frac{105}{16}
\]
Take the square root of both sides:
\[
x + \frac{5}{4} = \pm \sqrt{\frac{105}{16}}
\]
Simplify:
\[
x + \frac{5}{4} = \pm \frac{\sqrt{105}}{4}
\]
Solve for \(x\):
\[
x = -\frac{5}{4} \pm \frac{\sqrt{105}}{4}
\]
Combine terms:
\[
x = \frac{-5 \pm \sqrt{105}}{4}
\]
Approximate \(\sqrt{105} \approx 10.25\):
\[
x = \frac{-5 + 10.25}{4} \quad \text{or} \quad x = \frac{-5 - 10.25}{4}
\]
\[
x \approx \frac{5.25}{4} \quad \text{or} \quad x \approx \frac{-15.25}{4}
\]
\[
x \approx 1.3 \quad \text{or} \quad x \approx -3.8
\]

Solution: \(\boxed{1.3, -3.8}\)

---

#### 5) \(5x^2 = 5x + 1\)
Rewrite the equation:
\[
5x^2 - 5x - 1 = 0
\]
Divide the entire equation by 5:
\[
x^2 - x - \frac{1}{5} = 0
\]
Rewrite:
\[
x^2 - x = \frac{1}{5}
\]
Complete the square:
\[
x^2 - x + \left(\frac{1}{2}\right)^2 = \frac{1}{5} + \left(\frac{1}{2}\right)^2
\]
\[
x^2 - x + \frac{1}{4} = \frac{1}{5} + \frac{1}{4}
\]
\[
\left(x - \frac{1}{2}\right)^2 = \frac{4}{20} + \frac{5}{20}
\]
\[
\left(x - \frac{1}{2}\right)^2 = \frac{9}{20}
\]
Take the square root of both sides:
\[
x - \frac{1}{2} = \pm \sqrt{\frac{9}{20}}
\]
Simplify:
\[
x - \frac{1}{2} = \pm \frac{3}{\sqrt{20}}
\]
Rationalize the denominator:
\[
x - \frac{1}{2} = \pm \frac{3\sqrt{20}}{20} = \pm \frac{3 \cdot 2\sqrt{5}}{20} = \pm \frac{3\sqrt{5}}{10}
\]
Solve for \(x\):
\[
x = \frac{1}{2} \pm \frac{3\sqrt{5}}{10}
\]
Approximate \(\sqrt{5} \approx 2.24\):
\[
x = \frac{1}{2} + \frac{3 \cdot 2.24}{10} \quad \text{or} \quad x = \frac{1}{2} - \frac{3 \cdot 2.24}{10}
\]
\[
x \approx 0.5 + 0.672 \quad \text{or} \quad x \approx 0.5 - 0.672
\]
\[
x \approx 1.2 \quad \text{or} \quad x \approx -0.2
\]

Solution: \(\boxed{1.2, -0.2}\)

---

#### 6) \(2x^2 - 3x = 3\)
Rewrite the equation:
\[
2x^2 - 3x - 3 = 0
\]
Divide the entire equation by 2:
\[
x^2 - \frac{3}{2}x - \frac{3}{2} = 0
\]
Rewrite:
\[
x^2 - \frac{3}{2}x = \frac{3}{2}
\]
Complete the square:
\[
x^2 - \frac{3}{2}x + \left(\frac{3}{4}\right)^2 = \frac{3}{2} + \left(\frac{3}{4}\right)^2
\]
\[
x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{3}{2} + \frac{9}{16}
\]
\[
\left(x - \frac{3}{4}\right)^2 = \frac{24}{16} + \frac{9}{16}
\]
\[
\left(x - \frac{3}{4}\right)^2 = \frac{33}{16}
\]
Take the square root of both sides:
\[
x - \frac{3}{4} = \pm \sqrt{\frac{33}{16}}
\]
Simplify:
\[
x - \frac{3}{4} = \pm \frac{\sqrt{33}}{4}
\]
Solve for \(x\):
\[
x = \frac{3}{4} \pm \frac{\sqrt{33}}{4}
\]
Combine terms:
\[
x = \frac{3 \pm \sqrt{33}}{4}
\]
Approximate \(\sqrt{33} \approx 5.74\):
\[
x = \frac{3 + 5.74}{4} \quad \text{or} \quad x = \frac{3 - 5.74}{4}
\]
\[
x \approx \frac{8.74}{4} \quad \text{or} \quad x \approx \frac{-2.74}{4}
\]
\[
x \approx 2.2 \quad \text{or} \quad x \approx -0.7
\]

Solution: \(\boxed{2.2, -0.7}\)

---

Extension

#### A. The solutions to the equation \(2x^2 + ax + b = 0\) are \(x = 2 \pm \frac{\sqrt{11}}{2}\). Find the value of \(a\) and \(b\).

The given solutions are:
\[
x = 2 + \frac{\sqrt{11}}{2} \quad \text{and} \quad x = 2 - \frac{\sqrt{11}}{2}
\]
The quadratic equation can be written as:
\[
(x - (2 + \frac{\sqrt{11}}{2}))(x - (2 - \frac{\sqrt{11}}{2})) = 0
\]
Simplify:
\[
\left(x - 2 - \frac{\sqrt{11}}{2}\right)\left(x - 2 + \frac{\sqrt{11}}{2}\right) = 0
\]
Use the difference of squares:
\[
\left((x - 2) - \frac{\sqrt{11}}{2}\right)\left((x - 2) + \frac{\sqrt{11}}{2}\right) = (x - 2)^2 - \left(\frac{\sqrt{11}}{2}\right)^2
\]
\[
= (x - 2)^2 - \frac{11}{4}
\]
Expand \((x - 2)^2\):
\[
= x^2 - 4x + 4 - \frac{11}{4}
\]
Combine terms:
\[
= x^2 - 4x + \frac{16}{4} - \frac{11}{4}
\]
\[
= x^2 - 4x + \frac{5}{4}
\]
Multiply through by 4 to clear the fraction:
\[
4x^2 - 16x + 5 = 0
\]
Compare with \(2x^2 + ax + b = 0\). Divide the entire equation by 2:
\[
2x^2 - 8x + \frac{5}{2} = 0
\]
Thus:
\[
a = -8, \quad b = \frac{5}{2}
\]

Solution: \(\boxed{a = -8, b = \frac{5}{2}}\)

---

#### B. Starting with \(ax^2 + bx + c = 0\), prove the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Start with the general quadratic equation:
\[
ax^2 + bx + c = 0
\]
Divide through by \(a\):
\[
x^2 + \frac{b}{a}x + \frac{c}{a} = 0
\]
Rewrite:
\[
x^2 + \frac{b}{a}x = -\frac{c}{a}
\]
Complete the square:
\[
x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2
\]
\[
x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}
\]
\[
\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2}
\]
\[
\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}
\]
Take the square root of both sides:
\[
x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}
\]
Simplify:
\[
x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\]
Solve for \(x\):
\[
x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\]
Combine terms:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Solution: The quadratic formula is proven.

---

Final Answer:

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