Lecture on Solving Quadratic Equations by Extracting Square Roots

Introduction

• Perfect Squares: Numbers like 1, 4, 9, 16, 25, 36, etc.
• Objective: Solve quadratic equations by extracting square roots using perfect and non-perfect squares.

Solving Quadratic Equations

Perfect Squares

• Example 1: Solve (x^2 = 49)

  • Step 1: Take the square root of both sides.
  • (x = \pm \sqrt{49})
  • Solution: (x = \pm 7)

• Example 2: Solve (x^2 = 169)

  • Step 1: Take the square root of both sides.
  • (x = \pm \sqrt{169})
  • Solution: (x = \pm 13)

Non-Perfect Squares

• Example 3: Solve (x^2 = 75)

  • Step 1: Take the square root of both sides.
  • Factor 75 into (25 \times 3), where 25 is a perfect square.
  • (x = \pm (\sqrt{25} \times \sqrt{3}))
  • Solution: (x = \pm 5\sqrt{3})

• Example 4: Solve (x^2 = 80)

  • Step 1: Take the square root of both sides.
  • Factor 80 into (16 \times 5), where 16 is a perfect square.
  • (x = \pm (\sqrt{16} \times \sqrt{5}))
  • Solution: (x = \pm 4\sqrt{5})

Quadratic Equations with Expressions

• Example 5: Solve (2(x-5)^2 = 32)

  • Step 1: Divide both sides by 2.
  • ((x-5)^2 = 16)
  • Step 2: Take the square root of both sides.
  • (x-5 = \pm 4)
  • Solutions:
    • (x = 9)
    • (x = 1)

• Example 6: Solve (3(4x-1)^2 - 1 = 11)

  • Step 1: Add 1 to both sides.
  • (3(4x-1)^2 = 12)
  • Step 2: Divide by 3.
  • ((4x-1)^2 = 4)
  • Step 3: Take the square root.
  • (4x-1 = \pm 2)
  • Solutions:
    • (x = \frac{3}{4})
    • (x = -\frac{1}{4})

• Example 7: Solve ((2x-3)^2 = 18)

  • Step 1: Take the square root of both sides.
  • Factor 18 into (9 \times 2).
  • (2x-3 = \pm 3\sqrt{2})
  • Solutions:
    • (x = \frac{3 + 3\sqrt{2}}{2})
    • (x = \frac{3 - 3\sqrt{2}}{2})

• Example 8: Solve (2(5x+2)^2 = 64)

  • Step 1: Divide both sides by 2.
  • ((5x+2)^2 = 32)
  • Step 2: Take the square root.
  • Factor 32 into (16 \times 2).
  • (5x+2 = \pm 4\sqrt{2})
  • Solutions:
    • (x = \frac{-2 + 4\sqrt{2}}{5})
    • (x = \frac{-2 - 4\sqrt{2}}{5})

Conclusion

• Extracting square roots is a powerful method for solving quadratic equations.
• Perfect squares simplify the process, while non-perfect squares require factorization.
• Understanding how to manipulate and simplify expressions is key.

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