Skip to content

Pset.cts.01.sol

Completing the Square Solutions - Set 1

Problem 1.

Equation: \(x^2 - 6x + 5 = 0\)

Steps:

  1. Rewrite the equation: \(x^2 - 6x = -5\).
  2. Complete the square:
\[ x^2 - 6x + \left(\frac{6}{2}\right)^2 = -5 + \left(\frac{6}{2}\right)^2 \]
\[ x^2 - 6x + 9 = 4 \]
\[ (x - 3)^2 = 4 \]
  1. Solve for \(x\):
\[ x - 3 = \pm 2 \]
\[ x = 3 \pm 2 \]
\[ x = 5 \text{ or } x = 1 \]

Problem 2.

Equation: \(x^2 + 8x + 15 = 0\)

Steps:

  1. Rewrite the equation: \(x^2 + 8x = -15\).
  2. Complete the square:
\[ x^2 + 8x + \left(\frac{8}{2}\right)^2 = -15 + \left(\frac{8}{2}\right)^2 \]
\[ x^2 + 8x + 16 = 1 \]
\[ (x + 4)^2 = 1 \]
  1. Solve for \(x\):
\[ x + 4 = \pm 1 \]
\[ x = -4 \pm 1 \]
\[ x = -3 \text{ or } x = -5 \]

Problem 3.

Equation: \(2x^2 - 4x - 6 = 0\)

Steps: 1. Divide by 2: \(x^2 - 2x - 3 = 0\). 2. Rewrite the equation: \(x^2 - 2x = 3\). 3. Complete the square:

\[ x^2 - 2x + \left(\frac{2}{2}\right)^2 = 3 + \left(\frac{2}{2}\right)^2 \]
\[ x^2 - 2x + 1 = 4 \]
\[ (x - 1)^2 = 4 \]
  1. Solve for \(x\):
\[ x - 1 = \pm 2 \]
\[ x = 1 \pm 2 \]
\[ x = 3 \text{ or } x = -1 \]

Problem 4.

Equation: \(3x^2 - 12x + 7 = 0\)

Steps:

  1. Rewrite the equation: \(3x^2 - 12x = -7\).
  2. Divide by 3: \(x^2 - 4x = -\frac{7}{3}\).
  3. Complete the square:
\[ x^2 - 4x + \left(\frac{4}{2}\right)^2 = -\frac{7}{3} + \left(\frac{4}{2}\right)^2 \]
\[ x^2 - 4x + 4 = -\frac{7}{3} + \frac{16}{3} \]
\[ (x - 2)^2 = \frac{9}{3} \]
\[ (x - 2)^2 = 3 \]
  1. Solve for \(x\):
\[ x - 2 = \pm \sqrt{3} \]
\[ x = 2 \pm \sqrt{3} \]

Problem 5.

Equation: \(4x^2 + 20x + 25 = 0\)

Steps:

  1. Rewrite the equation: \(4x^2 + 20x = -25\).
  2. Divide by 4: \(x^2 + 5x = -\frac{25}{4}\).
  3. Complete the square: [ x^2 + 5x + \left(\frac{5}{2}\right)^2 = -\frac{25}{4} + \left(\frac{5}{2}\right)^2 ]
\[ x^2 + 5x + \frac{25}{4} = 0 \]
\[ \left(x + \frac{5}{2}\right)^2 = 0 \]
  1. Solve for \(x\):
\[ x + \frac{5}{2} = 0 \]
\[ x = -\frac{5}{2} \]

Problem 6.

Equation: \(x^2 + 6x - 7 = 0\)

Steps: 1. Rewrite the equation: \(x^2 + 6x = 7\). 2. Complete the square: [ x^2 + 6x + \left(\frac{6}{2}\right)^2 = 7 + \left(\frac{6}{2}\right)^2 ]

\[ x^2 + 6x + 9 = 16 \]
\[ (x + 3)^2 = 16 \]
  1. Solve for \(x\): [ x + 3 = \pm 4 ]
\[ x = -3 \pm 4 \]
\[ x = 1 \text{ or } x = -7 \]

Problem 7.

Equation: \(x^2 - 10x + 24 = 0\)

Steps:

  1. Rewrite the equation: \(x^2 - 10x = -24\).
  2. Complete the square:
\[ x^2 - 10x + \left(\frac{10}{2}\right)^2 = -24 + \left(\frac{10}{2}\right)^2 \]
\[ x^2 - 10x + 25 = 1 \]
\[ (x - 5)^2 = 1 \]
  1. Solve for \(x\):
\[ x - 5 = \pm 1 \]
\[ x = 5 \pm 1 \]
\[ x = 6 \text{ or } x = 4 \]

Problem 8.

Equation: \(5x^2 + 14x - 3 = 0\)

Steps: 1. Rewrite the equation: \(5x^2 + 14x = 3\). 2. Divide by 5: \(x^2 + \frac{14}{5}x = \frac{3}{5}\). 3. **Complete

the square**: [ x^2 + \frac{14}{5}x + \left(\frac{7}{5}\right)^2 = \frac{3}{5} + \left(\frac{7}{5}\right)^2 ]

\[ x^2 + \frac{14}{5}x + \frac{49}{25} = \frac{15 + 49}{25} \]
\[ \left(x + \frac{7}{5}\right)^2 = \frac{64}{25} \]
  1. Solve for \(x\): [ x + \frac{7}{5} = \pm \frac{8}{5} ]
\[ x = -\frac{7}{5} \pm \frac{8}{5} \]
\[ x = \frac{1}{5} \text{ or } x = -\frac{15}{5} = -3 \]

Problem 9.

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

Steps: 1. Rewrite the equation: \(x^2 + 2x = 8\). 2. Complete the square:

\[ x^2 + 2x + \left(\frac{2}{2}\right)^2 = 8 + \left(\frac{2}{2}\right)^2 \]
\[ x^2 + 2x + 1 = 9 \]
\[ (x + 1)^2 = 9 \]
  1. Solve for \(x\):

[ x + 1 = \pm 3 ] \ [ x = -1 \pm 3 ]

\[ x = 2 \text{ or } x = -4 \]

Problem 10.

Equation: \(6x^2 - 5x - 6 = 0\)

Steps:

  1. Rewrite the equation: \(6x^2 - 5x = 6\).
  2. Divide by 6: \(x^2 - \frac{5}{6}x = 1\).
  3. Complete the square: [ x^2 - \frac{5}{6}x + \left(\frac{5}{12}\right)^2 = 1 + \left(\frac{5}{12}\right)^2 ]
\[ x^2 - \frac{5}{6}x + \frac{25}{144} = 1 + \frac{25}{144} \]
\[ \left(x - \frac{5}{12}\right)^2 = \frac{169}{144} \]
  1. Solve for \(x\):
\[ x - \frac{5}{12} = \pm \frac{13}{12} \]
\[ x = \frac{5}{12} \pm \frac{13}{12} \]
\[ x = \frac{18}{12} = \frac{3}{2} \text{ or } x = -\frac{8}{12} = -\frac{2}{3} \]

Problem 11.

Equation: \(x^2 - 8x + 16 = 0\)

Steps:

  1. Rewrite the equation: \(x^2 - 8x = -16\).
  2. Complete the square:
\[ x^2 - 8x + \left(\frac{8}{2}\right)^2 = -16 + \left(\frac{8}{2}\right)^2 \]
\[ x^2 - 8x + 16 = 0 \]
\[ (x - 4)^2 = 0 \]
  1. Solve for \(x\):
\[ x - 4 = 0 \]
\[ x = 4 \]

Problem 12.

Equation: \(2x^2 + 4x - 3 = 0\)

Steps:

  1. Rewrite the equation: \(2x^2 + 4x = 3\).
  2. Divide by 2: \(x^2 + 2x = \frac{3}{2}\).
  3. Complete the square:
\[ x^2 + 2x + \left(\frac{2}{2}\right)^2 = \frac{3}{2} + \left(\frac{2}{2}\right)^2 \]
\[ x^2 + 2x + 1 = \frac{5}{2} \]
\[ (x + 1)^2 = \frac{5}{2} \]
  1. Solve for \(x\):
\[ x + 1 = \pm \sqrt{\frac{5}{2}} \]
\[ x = -1 \pm \sqrt{\frac{5}{2}} \]