Completing the Square - Mr-Mathematics.com - Free Printable

To solve the given quadratic equations by completing the square, we will follow a systematic approach. Let's go through each equation step by step.

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General Steps for Completing the Square:

1. Ensure the equation is in the form \( ax^2 + bx + c = 0 \).
2. If the coefficient of \( x^2 \) is not 1, divide the entire equation by \( a \).
3. Move the constant term \( c \) to the right side of the equation.
4. Add and subtract \( \left( \frac{b}{2} \right)^2 \) (the square of half the coefficient of \( x \)) to complete the square on the left side.
5. Rewrite the left side as a perfect square trinomial.
6. Solve for \( x \) by taking the square root of both sides and isolating \( x \).

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Solutions:

#### a) \( x^2 + 6x - 4 = 0 \)

1. Start with the equation:
\[
x^2 + 6x - 4 = 0
\]

2. Move the constant term \(-4\) to the right side:
\[
x^2 + 6x = 4
\]

3. To complete the square, add \( \left( \frac{6}{2} \right)^2 = 3^2 = 9 \) to both sides:
\[
x^2 + 6x + 9 = 4 + 9
\]

4. Rewrite the left side as a perfect square:
\[
(x + 3)^2 = 13
\]

5. Take the square root of both sides:
\[
x + 3 = \pm \sqrt{13}
\]

6. Solve for \( x \):
\[
x = -3 \pm \sqrt{13}
\]

Solution:
\[
\boxed{x = -3 \pm \sqrt{13}}
\]

---

#### b) \( x^2 + 8x + 3 = 0 \)

1. Start with the equation:
\[
x^2 + 8x + 3 = 0
\]

2. Move the constant term \(+3\) to the right side:
\[
x^2 + 8x = -3
\]

3. To complete the square, add \( \left( \frac{8}{2} \right)^2 = 4^2 = 16 \) to both sides:
\[
x^2 + 8x + 16 = -3 + 16
\]

4. Rewrite the left side as a perfect square:
\[
(x + 4)^2 = 13
\]

5. Take the square root of both sides:
\[
x + 4 = \pm \sqrt{13}
\]

6. Solve for \( x \):
\[
x = -4 \pm \sqrt{13}
\]

Solution:
\[
\boxed{x = -4 \pm \sqrt{13}}
\]

---

#### c) \( x^2 - 10x + 6 = 0 \)

1. Start with the equation:
\[
x^2 - 10x + 6 = 0
\]

2. Move the constant term \(+6\) to the right side:
\[
x^2 - 10x = -6
\]

3. To complete the square, add \( \left( \frac{-10}{2} \right)^2 = (-5)^2 = 25 \) to both sides:
\[
x^2 - 10x + 25 = -6 + 25
\]

4. Rewrite the left side as a perfect square:
\[
(x - 5)^2 = 19
\]

5. Take the square root of both sides:
\[
x - 5 = \pm \sqrt{19}
\]

6. Solve for \( x \):
\[
x = 5 \pm \sqrt{19}
\]

Solution:
\[
\boxed{x = 5 \pm \sqrt{19}}
\]

---

#### d) \( x^2 - 12x + 2 = 0 \)

1. Start with the equation:
\[
x^2 - 12x + 2 = 0
\]

2. Move the constant term \(+2\) to the right side:
\[
x^2 - 12x = -2
\]

3. To complete the square, add \( \left( \frac{-12}{2} \right)^2 = (-6)^2 = 36 \) to both sides:
\[
x^2 - 12x + 36 = -2 + 36
\]

4. Rewrite the left side as a perfect square:
\[
(x - 6)^2 = 34
\]

5. Take the square root of both sides:
\[
x - 6 = \pm \sqrt{34}
\]

6. Solve for \( x \):
\[
x = 6 \pm \sqrt{34}
\]

Solution:
\[
\boxed{x = 6 \pm \sqrt{34}}
\]

---

#### e) \( x^2 - 6x + 1 = 0 \)

1. Start with the equation:
\[
x^2 - 6x + 1 = 0
\]

2. Move the constant term \(+1\) to the right side:
\[
x^2 - 6x = -1
\]

3. To complete the square, add \( \left( \frac{-6}{2} \right)^2 = (-3)^2 = 9 \) to both sides:
\[
x^2 - 6x + 9 = -1 + 9
\]

4. Rewrite the left side as a perfect square:
\[
(x - 3)^2 = 8
\]

5. Take the square root of both sides:
\[
x - 3 = \pm \sqrt{8}
\]

6. Simplify \( \sqrt{8} = 2\sqrt{2} \) and solve for \( x \):
\[
x = 3 \pm 2\sqrt{2}
\]

Solution:
\[
\boxed{x = 3 \pm 2\sqrt{2}}
\]

---

#### f) \( x^2 + 2x = 2 \)

1. Start with the equation:
\[
x^2 + 2x = 2
\]

2. To complete the square, add \( \left( \frac{2}{2} \right)^2 = 1^2 = 1 \) to both sides:
\[
x^2 + 2x + 1 = 2 + 1
\]

3. Rewrite the left side as a perfect square:
\[
(x + 1)^2 = 3
\]

4. Take the square root of both sides:
\[
x + 1 = \pm \sqrt{3}
\]

5. Solve for \( x \):
\[
x = -1 \pm \sqrt{3}
\]

Solution:
\[
\boxed{x = -1 \pm \sqrt{3}}
\]

---

#### g) \( x^2 + 3 = 5x \)

1. Rearrange the equation to standard form:
\[
x^2 - 5x + 3 = 0
\]

2. Move the constant term \(+3\) to the right side:
\[
x^2 - 5x = -3
\]

3. To complete the square, add \( \left( \frac{-5}{2} \right)^2 = \left( -\frac{5}{2} \right)^2 = \frac{25}{4} \) to both sides:
\[
x^2 - 5x + \frac{25}{4} = -3 + \frac{25}{4}
\]

4. Simplify the right side:
\[
-3 = -\frac{12}{4}, \quad \text{so } -\frac{12}{4} + \frac{25}{4} = \frac{13}{4}
\]

5. Rewrite the left side as a perfect square:
\[
\left( x - \frac{5}{2} \right)^2 = \frac{13}{4}
\]

6. Take the square root of both sides:
\[
x - \frac{5}{2} = \pm \sqrt{\frac{13}{4}}
\]

7. Simplify \( \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2} \) and solve for \( x \):
\[
x = \frac{5}{2} \pm \frac{\sqrt{13}}{2}
\]

8. Combine the terms:
\[
x = \frac{5 \pm \sqrt{13}}{2}
\]

Solution:
\[
\boxed{x = \frac{5 \pm \sqrt{13}}{2}}
\]

---

#### h) \( x^2 = 5 - 9x \)

1. Rearrange the equation to standard form:
\[
x^2 + 9x - 5 = 0
\]

2. Move the constant term \(-5\) to the right side:
\[
x^2 + 9x = 5
\]

3. To complete the square, add \( \left( \frac{9}{2} \right)^2 = \left( \frac{9}{2} \right)^2 = \frac{81}{4} \) to both sides:
\[
x^2 + 9x + \frac{81}{4} = 5 + \frac{81}{4}
\]

4. Simplify the right side:
\[
5 = \frac{20}{4}, \quad \text{so } \frac{20}{4} + \frac{81}{4} = \frac{101}{4}
\]

5. Rewrite the left side as a perfect square:
\[
\left( x + \frac{9}{2} \right)^2 = \frac{101}{4}
\]

6. Take the square root of both sides:
\[
x + \frac{9}{2} = \pm \sqrt{\frac{101}{4}}
\]

7. Simplify \( \sqrt{\frac{101}{4}} = \frac{\sqrt{101}}{2} \) and solve for \( x \):
\[
x = -\frac{9}{2} \pm \frac{\sqrt{101}}{2}
\]

8. Combine the terms:
\[
x = \frac{-9 \pm \sqrt{101}}{2}
\]

Solution:
\[
\boxed{x = \frac{-9 \pm \sqrt{101}}{2}}
\]

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Final Answers:

\[
\boxed{
\begin{aligned}
&\text{a) } x = -3 \pm \sqrt{13} \\
&\text{b) } x = -4 \pm \sqrt{13} \\
&\text{c) } x = 5 \pm \sqrt{19} \\
&\text{d) } x = 6 \pm \sqrt{34} \\
&\text{e) } x = 3 \pm 2\sqrt{2} \\
&\text{f) } x = -1 \pm \sqrt{3} \\
&\text{g) } x = \frac{5 \pm \sqrt{13}}{2} \\
&\text{h) } x = \frac{-9 \pm \sqrt{101}}{2}
\end{aligned}
}
\]