Algebra I - Unit 7C: Efficient Solution Methods

Algebra I - Unit 7C: Which Method is Most Efficient?

Directions

• Check all methods that can be used to find EXACT solutions of the polynomial equations.
• Circle the method that you chose - use each method AT LEAST ONCE.
• Find all solutions - leave your answers EXACT.

Problem 1: $3x^2 - 27x = 0$

• Methods Applicable: Factoring, Completing the Square, Quadratic Formula
• Chosen Method: Factoring
• Solution:
  • Factor out $3x$: $3x(x - 9) = 0$
  • Set each factor to zero: $3x = 0$ or $x - 9 = 0$
  • Solve for $x$: $x = 0$ or $x = 9$
  • Solutions: $x = 0, 9$

Problem 2: $-x^2 + 5x - 4 = 0$

• Methods Applicable: Factoring, Completing the Square, Quadratic Formula
• Chosen Method: Factoring
• Solution:
  • Multiply by -1: $x^2 - 5x + 4 = 0$
  • Factor: $(x - 1)(x - 4) = 0$
  • Set each factor to zero: $x - 1 = 0$ or $x - 4 = 0$
  • Solve for $x$: $x = 1$ or $x = 4$
  • Solutions: $x = 1, 4$

Problem 3: $3x^2 + 2x - 3 = 0$

• Methods Applicable: Completing the Square, Quadratic Formula
• Chosen Method: Quadratic Formula
• Solution:
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Where $a = 3$, $b = 2$, and $c = -3$
  • $x = \frac{-2 \pm \sqrt{2^2 - 4(3)(-3)}}{2(3)}$
  • $x = \frac{-2 \pm \sqrt{4 + 36}}{6}$
  • $x = \frac{-2 \pm \sqrt{40}}{6}$
  • $x = \frac{-2 \pm 2\sqrt{10}}{6}$
  • $x = \frac{-1 \pm \sqrt{10}}{3}$
  • Solutions: $x = \frac{-1 \pm \sqrt{10}}{3}$

Problem 4: $2x^2 + 7x = 15$

• Methods Applicable: Factoring, Completing the Square, Quadratic Formula
• Chosen Method: Factoring
• Solution:
  • Rearrange: $2x^2 + 7x - 15 = 0$
  • Factor: $(2x - 3)(x + 5) = 0$
  • Set each factor to zero: $2x - 3 = 0$ or $x + 5 = 0$
  • Solve for $x$: $x = \frac{3}{2}$ or $x = -5$
  • Solutions: $x = \frac{3}{2}, -5$

Problem 5: $7(x + 2)^2 = 56$

• Methods Applicable: Inverse Operations, Completing the Square, Quadratic Formula
• Chosen Method: Inverse Operations
• Solution:
  • Divide by 7: $(x + 2)^2 = 8$
  • Take the square root: $x + 2 = \pm \sqrt{8}$
  • Simplify: $x + 2 = \pm 2\sqrt{2}$
  • Solve for $x$: $x = -2 \pm 2\sqrt{2}$
  • Solutions: $x = -2 \pm 2\sqrt{2}$

Problem 6: $4x(x + 1) = 3$

• Methods Applicable: Completing the Square, Quadratic Formula
• Chosen Method: Quadratic Formula
• Solution:
  • Expand and rearrange: $4x^2 + 4x - 3 = 0$
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Where $a = 4$, $b = 4$, and $c = -3$
  • $x = \frac{-4 \pm \sqrt{4^2 - 4(4)(-3)}}{2(4)}$
  • $x = \frac{-4 \pm \sqrt{16 + 48}}{8}$
  • $x = \frac{-4 \pm \sqrt{64}}{8}$
  • $x = \frac{-4 \pm 8}{8}$
  • $x = \frac{-4 + 8}{8}$ or $x = \frac{-4 - 8}{8}$
  • $x = \frac{4}{8}$ or $x = \frac{-12}{8}$
  • Solutions: $x = \frac{1}{2}, \frac{-3}{2}$

Problem 7: $4x^2 + 32 = -16$

• Methods Applicable: Inverse Operations, Completing the Square, Quadratic Formula
• Chosen Method: Inverse Operations
• Solution:
  • Subtract 32: $4x^2 = -48$
  • Divide by 4: $x^2 = -12$
  • Take the square root: $x = \pm \sqrt{-12}$
  • Solutions: No Real Solution

Problem 8: $81x^2 + 72x + 16 = 0$

• Methods Applicable: Factoring, Completing the Square, Quadratic Formula
• Chosen Method: Factoring
• Solution:
  • Factor: $(9x + 4)^2 = 0$
  • Set the factor to zero: $9x + 4 = 0$
  • Solve for $x$: $x = -\frac{4}{9}$
  • Solutions: $x = -\frac{4}{9}$

Problem 9: $5x^2 + 4x + 3 = 0$

• Methods Applicable: Completing the Square, Quadratic Formula
• Chosen Method: Quadratic Formula
• Solution:
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Where $a = 5$, $b = 4$, and $c = 3$
  • $x = \frac{-4 \pm \sqrt{4^2 - 4(5)(3)}}{2(5)}$
  • $x = \frac{-4 \pm \sqrt{16 - 60}}{10}$
  • $x = \frac{-4 \pm \sqrt{-44}}{10}$
  • Solutions: No Real Solution

Problem 10: $x^2 - 18x = -21$

• Methods Applicable: Completing the Square, Quadratic Formula
• Chosen Method: Completing the Square
• Solution:
  • Rearrange: $x^2 - 18x + 21 = 0$
  • Complete the square: $x^2 - 18x + 81 = -21 + 81$
  • $(x - 9)^2 = 60$
  • Take the square root: $x - 9 = \pm \sqrt{60}$
  • Simplify: $x - 9 = \pm 2\sqrt{15}$
  • Solve for $x$: $x = 9 \pm 2\sqrt{15}$
  • Solutions: $x = 9 \pm 2\sqrt{15}$

Problem 11: $x(x - 3) = 7$

• Methods Applicable: Completing the Square, Quadratic Formula
• Chosen Method: Quadratic Formula
• Solution:
  • Expand and rearrange: $x^2 - 3x - 7 = 0$
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Where $a = 1$, $b = -3$, and $c = -7$
  • $x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-7)}}{2(1)}$
  • $x = \frac{3 \pm \sqrt{9 + 28}}{2}$
  • $x = \frac{3 \pm \sqrt{37}}{2}$
  • Solutions: $x = \frac{3 \pm \sqrt{37}}{2}$

Problem 12: $x^2 + 7x + 8 = 6$

• Methods Applicable: Factoring, Completing the Square, Quadratic Formula
• Chosen Method: Factoring
• Solution:
  • Rearrange: $x^2 + 7x + 2 = 0$
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Where $a = 1$, $b = 7$, and $c = 2$
  • $x = \frac{-7 \pm \sqrt{7^2 - 4(1)(2)}}{2(1)}$
  • $x = \frac{-7 \pm \sqrt{49 - 8}}{2}$
  • $x = \frac{-7 \pm \sqrt{41}}{2}$
  • Solutions: $x = \frac{-7 \pm \sqrt{41}}{2}$