Solving Quadratic Equations by Completing the Square - Link ... - Free Printable

Problem Analysis:

The task involves matching quadratic equations (in the "Left" column) with their correct solutions (in the "Right" column). To solve this, we will use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where the quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).

We will calculate the solutions for each quadratic equation and match them with the given solutions in the "Right" column.

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Step-by-Step Solution:

#### Equation A: \( x^2 - 8x + 2 = 0 \)

- Here, \( a = 1 \), \( b = -8 \), and \( c = 2 \).
- Using the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(2)}}{2(1)}
\]
\[
x = \frac{8 \pm \sqrt{64 - 8}}{2}
\]
\[
x = \frac{8 \pm \sqrt{56}}{2}
\]
\[
x = \frac{8 \pm 2\sqrt{14}}{2}
\]
\[
x = 4 \pm \sqrt{14}
\]

- Solution: \( x = 4 \pm \sqrt{14} \)

This does not match any of the given solutions in the "Right" column. There seems to be an error in the problem statement or the provided solutions.

---

#### Equation B: \( x^2 + 6x - 1 = 0 \)

- Here, \( a = 1 \), \( b = 6 \), and \( c = -1 \).
- Using the quadratic formula:
\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-1)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 + 4}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{40}}{2}
\]
\[
x = \frac{-6 \pm 2\sqrt{10}}{2}
\]
\[
x = -3 \pm \sqrt{10}
\]

- Solution: \( x = -3 \pm \sqrt{10} \)

This does not match any of the given solutions in the "Right" column. There seems to be an error in the problem statement or the provided solutions.

---

#### Equation C: \( x^2 - 8x - 3 = 0 \)

- Here, \( a = 1 \), \( b = -8 \), and \( c = -3 \).
- Using the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-3)}}{2(1)}
\]
\[
x = \frac{8 \pm \sqrt{64 + 12}}{2}
\]
\[
x = \frac{8 \pm \sqrt{76}}{2}
\]
\[
x = \frac{8 \pm 2\sqrt{19}}{2}
\]
\[
x = 4 \pm \sqrt{19}
\]

- Solution: \( x = 4 \pm \sqrt{19} \)

This matches the solution in row A of the "Right" column.

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#### Equation D: \( x^2 + 6x + 4 = 0 \)

- Here, \( a = 1 \), \( b = 6 \), and \( c = 4 \).
- Using the quadratic formula:
\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(4)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 - 16}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{20}}{2}
\]
\[
x = \frac{-6 \pm 2\sqrt{5}}{2}
\]
\[
x = -3 \pm \sqrt{5}
\]

- Solution: \( x = -3 \pm \sqrt{5} \)

This does not match any of the given solutions in the "Right" column. There seems to be an error in the problem statement or the provided solutions.

---

#### Equation E: \( x^2 - 4x + 1 = 0 \)

- Here, \( a = 1 \), \( b = -4 \), and \( c = 1 \).
- Using the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}
\]
\[
x = \frac{4 \pm \sqrt{16 - 4}}{2}
\]
\[
x = \frac{4 \pm \sqrt{12}}{2}
\]
\[
x = \frac{4 \pm 2\sqrt{3}}{2}
\]
\[
x = 2 \pm \sqrt{3}
\]

- Solution: \( x = 2 \pm \sqrt{3} \)

This matches the solution in row B of the "Right" column.

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Final Matching:

- Equation A: \( x^2 - 8x + 2 = 0 \) → No match (error in problem or solutions)
- Equation B: \( x^2 + 6x - 1 = 0 \) → No match (error in problem or solutions)
- Equation C: \( x^2 - 8x - 3 = 0 \) → Matches row A: \( x = 4 \pm \sqrt{19} \)
- Equation D: \( x^2 + 6x + 4 = 0 \) → No match (error in problem or solutions)
- Equation E: \( x^2 - 4x + 1 = 0 \) → Matches row B: \( x = 2 \pm \sqrt{3} \)

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Correct Matches:

\[
\boxed{C \to A, E \to B}
\]