Completing the Square worksheet with problems and solutions.

Here are the solutions to the problems on the worksheet, broken down by section.

Section A

*Solve the following equations.*

1) $(x + 7)^2 - 9 = 0$
* Add 9 to both sides: $(x + 7)^2 = 9$
* Take the square root of both sides: $x + 7 = \pm 3$
* Subtract 7 from both sides: $x = -7 \pm 3$
* Answer: $x = -4$ or $x = -10$

2) $(x - 4)^2 - 25 = 0$
* Add 25 to both sides: $(x - 4)^2 = 25$
* Take the square root: $x - 4 = \pm 5$
* Add 4 to both sides: $x = 4 \pm 5$
* Answer: $x = 9$ or $x = -1$

3) $2(x + 5)^2 - 32 = 0$
* Add 32 to both sides: $2(x + 5)^2 = 32$
* Divide by 2: $(x + 5)^2 = 16$
* Take the square root: $x + 5 = \pm 4$
* Subtract 5: $x = -5 \pm 4$
* Answer: $x = -1$ or $x = -9$

4) $5(x + 4)^2 - 180 = 0$
* Add 180 to both sides: $5(x + 4)^2 = 180$
* Divide by 5: $(x + 4)^2 = 36$
* Take the square root: $x + 4 = \pm 6$
* Subtract 4: $x = -4 \pm 6$
* Answer: $x = 2$ or $x = -10$

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Section B

*Solve by completing the square. Leave answers in surd form.*

1) $x^2 + 4x - 30 = 0$
* Move constant to right: $x^2 + 4x = 30$
* Halve the middle coefficient (4/2 = 2) and square it ($2^2 = 4$). Add to both sides.
* $x^2 + 4x + 4 = 34$
* Factor left side: $(x + 2)^2 = 34$
* Square root: $x + 2 = \pm\sqrt{34}$
* Answer: $x = -2 \pm \sqrt{34}$

2) $x^2 + 18x - 21 = 0$
* Move constant: $x^2 + 18x = 21$
* Halve 18 (9) and square it (81). Add to both sides.
* $x^2 + 18x + 81 = 21 + 81$
* $(x + 9)^2 = 102$
* Answer: $x = -9 \pm \sqrt{102}$

3) $x^2 - 6x - 3 = 0$
* Move constant: $x^2 - 6x = 3$
* Halve -6 (-3) and square it (9). Add to both sides.
* $x^2 - 6x + 9 = 3 + 9$
* $(x - 3)^2 = 12$
* Square root: $x - 3 = \pm\sqrt{12}$. Simplify $\sqrt{12}$ to $2\sqrt{3}$.
* Answer: $x = 3 \pm 2\sqrt{3}$

4) $x^2 - x - 7 = 0$
* Move constant: $x^2 - x = 7$
* Halve -1 ($-\frac{1}{2}$) and square it ($\frac{1}{4}$). Add to both sides.
* $x^2 - x + \frac{1}{4} = 7 + \frac{1}{4} = \frac{29}{4}$
* $(x - \frac{1}{2})^2 = \frac{29}{4}$
* Square root: $x - \frac{1}{2} = \pm\frac{\sqrt{29}}{2}$
* Answer: $x = \frac{1 \pm \sqrt{29}}{2}$

5) $x^2 + 12x - 5 = 23$
* Combine constants first: $x^2 + 12x = 28$
* Halve 12 (6) and square it (36). Add to both sides.
* $x^2 + 12x + 36 = 28 + 36$
* $(x + 6)^2 = 64$
* Square root: $x + 6 = \pm 8$
* $x = -6 \pm 8$
* Answer: $x = 2$ or $x = -14$

6) $x^2 - 32x + 45 = -36$
* Combine constants: $x^2 - 32x = -81$
* Halve -32 (-16) and square it (256). Add to both sides.
* $x^2 - 32x + 256 = -81 + 256$
* $(x - 16)^2 = 175$
* Square root: $x - 16 = \pm\sqrt{175}$. Simplify $\sqrt{175}$ to $5\sqrt{7}$.
* Answer: $x = 16 \pm 5\sqrt{7}$

7) $x^2 + 3x - 24 = 1$
* Combine constants: $x^2 + 3x = 25$
* Halve 3 ($\frac{3}{2}$) and square it ($\frac{9}{4}$). Add to both sides.
* $x^2 + 3x + \frac{9}{4} = 25 + \frac{9}{4} = \frac{109}{4}$
* $(x + \frac{3}{2})^2 = \frac{109}{4}$
* Square root: $x + \frac{3}{2} = \pm\frac{\sqrt{109}}{2}$
* Answer: $x = \frac{-3 \pm \sqrt{109}}{2}$

8) $x^2 - 5x - 4 = 5$
* Combine constants: $x^2 - 5x = 9$
* Halve -5 ($-\frac{5}{2}$) and square it ($\frac{25}{4}$). Add to both sides.
* $x^2 - 5x + \frac{25}{4} = 9 + \frac{25}{4} = \frac{61}{4}$
* $(x - \frac{5}{2})^2 = \frac{61}{4}$
* Square root: $x - \frac{5}{2} = \pm\frac{\sqrt{61}}{2}$
* Answer: $x = \frac{5 \pm \sqrt{61}}{2}$

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Section C

*Solve by completing the square. Answers to one decimal place.*

1) $2x^2 + 4x - 18 = 0$
* Divide entire equation by 2: $x^2 + 2x - 9 = 0$
* Move constant: $x^2 + 2x = 9$
* Halve 2 (1) and square it (1). Add to both sides.
* $(x + 1)^2 = 10$
* $x + 1 = \pm\sqrt{10}$
* $x = -1 \pm 3.16...$
* Answer: $x \approx 2.2$ or $x \approx -4.2$

2) $2x^2 + 8x - 5 = 0$
* Divide by 2: $x^2 + 4x - 2.5 = 0$
* Move constant: $x^2 + 4x = 2.5$
* Halve 4 (2) and square it (4). Add to both sides.
* $(x + 2)^2 = 6.5$
* $x + 2 = \pm\sqrt{6.5}$
* $x = -2 \pm 2.55...$
* Answer: $x \approx 0.6$ or $x \approx -4.6$

3) $3x^2 - 12x + 4 = 0$
* Divide by 3: $x^2 - 4x + \frac{4}{3} = 0$
* Move constant: $x^2 - 4x = -\frac{4}{3}$
* Halve -4 (-2) and square it (4). Add to both sides.
* $(x - 2)^2 = 4 - \frac{4}{3} = \frac{8}{3}$
* $x - 2 = \pm\sqrt{\frac{8}{3}}$
* $x = 2 \pm 1.63...$
* Answer: $x \approx 3.6$ or $x \approx 0.4$

4) $2x^2 + 5x - 10 = 0$
* Divide by 2: $x^2 + 2.5x - 5 = 0$
* Move constant: $x^2 + 2.5x = 5$
* Halve 2.5 (1.25) and square it (1.5625). Add to both sides.
* $(x + 1.25)^2 = 6.5625$
* $x + 1.25 = \pm\sqrt{6.5625}$
* $x = -1.25 \pm 2.56...$
* Answer: $x \approx 1.3$ or $x \approx -3.8$

5) $5x^2 = 5x + 1$
* Rearrange: $5x^2 - 5x - 1 = 0$
* Divide by 5: $x^2 - x - 0.2 = 0$
* Move constant: $x^2 - x = 0.2$
* Halve -1 (-0.5) and square it (0.25). Add to both sides.
* $(x - 0.5)^2 = 0.45$
* $x - 0.5 = \pm\sqrt{0.45}$
* $x = 0.5 \pm 0.67...$
* Answer: $x \approx 1.2$ or $x \approx -0.2$

6) $2x^2 - 3x = 3$
* Divide by 2: $x^2 - 1.5x = 1.5$
* Halve -1.5 (-0.75) and square it (0.5625). Add to both sides.
* $(x - 0.75)^2 = 1.5 + 0.5625 = 2.0625$
* $x - 0.75 = \pm\sqrt{2.0625}$
* $x = 0.75 \pm 1.436...$
* Answer: $x \approx 2.2$ or $x \approx -0.7$

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Extension

A. Find the value of a and b.
Equation: $2x^2 + ax + b = 0$
Solutions given: $x = 2 + \sqrt{\frac{11}{2}}$

Let's solve $2x^2 + ax + b = 0$ by completing the square to match the form of the solution.
1. Divide by 2: $x^2 + \frac{a}{2}x + \frac{b}{2} = 0$
2. Move constant: $x^2 + \frac{a}{2}x = -\frac{b}{2}$
3. Complete square: The term added is $(\frac{a}{4})^2 = \frac{a^2}{16}$.
$(x + \frac{a}{4})^2 = -\frac{b}{2} + \frac{a^2}{16}$
4. Take square root: $x + \frac{a}{4} = \pm \sqrt{\frac{a^2}{16} - \frac{b}{2}}$
5. Isolate x: $x = -\frac{a}{4} \pm \sqrt{\frac{a^2 - 8b}{16}}$
$x = -\frac{a}{4} \pm \frac{1}{4}\sqrt{a^2 - 8b}$

Compare this to the given solution: $x = 2 \pm \sqrt{\frac{11}{2}}$

First, look at the integer part outside the square root:
$-\frac{a}{4} = 2 \implies a = -8$

Now substitute $a = -8$ into the square root part to find $b$:
The term inside the square root in our derived formula is $\frac{a^2 - 8b}{16}$.
The term inside the square root in the given answer is $\frac{11}{2}$.
So, $\frac{(-8)^2 - 8b}{16} = \frac{11}{2}$
$\frac{64 - 8b}{16} = \frac{11}{2}$
Multiply both sides by 16:
$64 - 8b = 88$
$-8b = 24$
$b = -3$

Answer: $a = -8$, $b = -3$

B. Prove the quadratic formula.
Starting with $ax^2 + bx + c = 0$:

1. Subtract $c$ from both sides:
$ax^2 + bx = -c$

2. Divide every term by $a$:
$x^2 + \frac{b}{a}x = -\frac{c}{a}$

3. To complete the square, take half of the coefficient of $x$ (which is $\frac{b}{a}$), so we get $\frac{b}{2a}$. Square it to get $\frac{b^2}{4a^2}$. Add this to both sides:
$x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}$

4. Factor the left side as a perfect square:
$(x + \frac{b}{2a})^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$

5. Find a common denominator for the right side ($4a^2$):
$-\frac{c}{a} = -\frac{4ac}{4a^2}$
So, $(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$

6. Take the square root of both sides:
$x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}$
$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$

7. Subtract $\frac{b}{2a}$ from both sides to isolate $x$:
$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$

8. Combine the fractions:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

(Proof Complete)