Solving Quadratic Equations by Extracting Square Roots

Perfect Squares and Extracting Square Roots

Introduction

  • Familiarization with perfect squares (1, 4, 9, 16, 25, 36, …).
  • Solving quadratic equations by extracting square roots.

Extracting Square Roots When Given is a Perfect Square

  • Example: x2=49x^2 = 49
  • 49 is a perfect square.
  • Take the square root of both sides:
    x2=±49\sqrt{x^2} = \pm \sqrt{49}
  • Simplify:
    x=±7x = \pm 7
  • Solutions: x=7x = 7 or x=−7x = -7

Additional Example with a Perfect Square

  • Equation: x2=169x^2 = 169
  • 169 is a perfect square.
  • Take the square root of both sides:
    x2=±169\sqrt{x^2} = \pm \sqrt{169}
  • Simplify:
    x=±13x = \pm 13
  • Solutions: x=13x = 13 or x=−13x = -13

Extracting Square Roots When Given is Not a Perfect Square

  • Example: x2=75x^2 = 75
  • 75 is not a perfect square.
  • Take the square root of both sides:
    x2=±75\sqrt{x^2} = \pm \sqrt{75}
  • Factor 75 into 25×325 \times 3, where 25 is a perfect square.
  • Rewrite:
    x=±25×3=±25×3x = \pm \sqrt{25 \times 3} = \pm \sqrt{25} \times \sqrt{3}
  • Simplify:
    x=±53x = \pm 5\sqrt{3}

Another Example with a Non-Perfect Square

  • Equation: x2=80x^2 = 80
  • 80 is not a perfect square.
  • Take the square root of both sides:
    x2=±80\sqrt{x^2} = \pm \sqrt{80}
  • Factor 80 into 16×516 \times 5, where 16 is a perfect square.
  • Rewrite:
    x=±16×5=±16×5x = \pm \sqrt{16 \times 5} = \pm \sqrt{16} \times \sqrt{5}
  • Simplify:
    x=±45x = \pm 4\sqrt{5}

Example with Additional Steps

  • Equation: 2(x−5)2=322(x - 5)^2 = 32
  • Divide both sides by 2:
    (x−5)2=16(x - 5)^2 = 16
  • Take the square root of both sides:
    (x−5)2=±16\sqrt{(x - 5)^2} = \pm \sqrt{16}
  • Simplify:
    x−5=±4x - 5 = \pm 4
  • Create two equations:
    • x−5=4x - 5 = 4
    • x−5=−4x - 5 = -4
  • Solve for x:
    • x=4+5=9x = 4 + 5 = 9
    • x=−4+5=1x = -4 + 5 = 1
  • Solutions: x=9x = 9 or x=1x = 1

More Complex Example

  • Equation: 3(4x−1)2−1=113(4x - 1)^2 - 1 = 11
  • Add 1 to both sides:
    3(4x−1)2=123(4x - 1)^2 = 12
  • Divide both sides by 3:
    (4x−1)2=4(4x - 1)^2 = 4
  • Take the square root of both sides:
    (4x−1)2=±4\sqrt{(4x - 1)^2} = \pm \sqrt{4}
  • Simplify:
    4x−1=±24x - 1 = \pm 2
  • Create two equations:
    • 4x−1=24x - 1 = 2
    • 4x−1=−24x - 1 = -2
  • Solve for x:
    • 4x=3  ⟹  x=344x = 3 \implies x = \frac{3}{4}
    • 4x=−1  ⟹  x=−144x = -1 \implies x = -\frac{1}{4}

Extracting Square Roots with Non-Perfect Square Results

  • Equation: (2x−3)2=18(2x - 3)^2 = 18
  • Take the square root of both sides:
    (2x−3)2=±18\sqrt{(2x - 3)^2} = \pm \sqrt{18}
  • Factor 18 into 9×29 \times 2, where 9 is a perfect square.
  • Rewrite:
    2x−3=±9×2=±9×22x - 3 = \pm \sqrt{9 \times 2} = \pm \sqrt{9} \times \sqrt{2}
  • Simplify:
    2x−3=±322x - 3 = \pm 3\sqrt{2}
  • Create two equations:
    • 2x−3=322x - 3 = 3\sqrt{2}
    • 2x−3=−322x - 3 = -3\sqrt{2}
  • Solve for x:
    • 2x=3+32  ⟹  x=3+3222x = 3 + 3\sqrt{2} \implies x = \frac{3 + 3\sqrt{2}}{2}
    • 2x=3−32  ⟹  x=3−3222x = 3 - 3\sqrt{2} \implies x = \frac{3 - 3\sqrt{2}}{2}

Another Example with Non-Perfect Square Results

  • Equation: 2(5x+2)2=642(5x + 2)^2 = 64
  • Divide both sides by 2:
    (5x+2)2=32(5x + 2)^2 = 32
  • Take the square root of both sides:
    (5x+2)2=±32\sqrt{(5x + 2)^2} = \pm \sqrt{32}
  • Factor 32 into 16×216 \times 2, where 16 is a perfect square.
  • Rewrite:
    5x+2=±16×2=±16×25x + 2 = \pm \sqrt{16 \times 2} = \pm \sqrt{16} \times \sqrt{2}
  • Simplify:
    5x+2=±425x + 2 = \pm 4\sqrt{2}
  • Create two equations:
    • 5x+2=425x + 2 = 4\sqrt{2}
    • 5x+2=−425x + 2 = -4\sqrt{2}
  • Solve for x:
    • 5x=−2+42  ⟹  x=−2+4255x = -2 + 4\sqrt{2} \implies x = \frac{-2 + 4\sqrt{2}}{5}
    • 5x=−2−42  ⟹  x=−2−4255x = -2 - 4\sqrt{2} \implies x = \frac{-2 - 4\sqrt{2}}{5}