Math Problem Statement

solve the quadratics

Solve the quadratic equations from the image.

Solution

Let me extract the equations from the image and solve the quadratics. I'll break them down one by one.

The quadratic equations from the image are:

  1. c)x(x+5)=6c) \: x(x+5) = 6
  2. d)x(x+19)=72d) \: x(x+19) = 72
  3. e)3x23x2=0e) \: 3x^2 - 3x - 2 = 0
  4. f)3x28x+16=0f) \: 3x^2 - 8x + 16 = 0
  5. g)5x226x+24=0g) \: 5x^2 - 26x + 24 = 0
  6. h)x(x+12)3x=2h) \: x(x + 12) - 3x = 2
  7. i)5x28=0i) \: 5x^2 - 8 = 0
  8. j)4x212x7=0j) \: 4x^2 - 12x - 7 = 0
  9. k)x22x8=0k) \: x^2 - 2x - 8 = 0
  10. l)6x2=10x+14l) \: 6x^2 = 10x + 14
  11. m)7x224=2(3x12)m) \: 7x^2 - 24 = 2(3x - 12)
  12. n)x2+5x28=0n) \: x^2 + 5x - 28 = 0

I'll start solving each equation systematically.Here are the solutions to the quadratic equations:

  1. c)x(x+5)=6c) \: x(x + 5) = 6

    • Solutions: x=6x = -6, x=1x = 1
  2. d)x(x+19)=72d) \: x(x + 19) = 72

    • Solutions: x=19+6492x = \frac{-19 + \sqrt{649}}{2}, x=196492x = \frac{-19 - \sqrt{649}}{2}
  3. e)3x23x2=0e) \: 3x^2 - 3x - 2 = 0

    • Solutions: x=1336x = \frac{1 - \sqrt{33}}{6}, x=1+336x = \frac{1 + \sqrt{33}}{6}
  4. f)3x28x+16=0f) \: 3x^2 - 8x + 16 = 0

    • Solutions: Complex roots x=434i23x = \frac{4}{3} - \frac{4i\sqrt{2}}{3}, x=43+4i23x = \frac{4}{3} + \frac{4i\sqrt{2}}{3}
  5. g)5x226x+24=0g) \: 5x^2 - 26x + 24 = 0

    • Solutions: x=65x = \frac{6}{5}, x=4x = 4
  6. h)x(x+12)3x=2h) \: x(x + 12) - 3x = 2

    • Solutions: x=9+892x = \frac{-9 + \sqrt{89}}{2}, x=9892x = \frac{-9 - \sqrt{89}}{2}
  7. i)5x28=0i) \: 5x^2 - 8 = 0

    • Solutions: x=2105x = \frac{-2\sqrt{10}}{5}, x=2105x = \frac{2\sqrt{10}}{5}
  8. j)4x212x7=0j) \: 4x^2 - 12x - 7 = 0

    • Solutions: x=12x = \frac{-1}{2}, x=72x = \frac{7}{2}
  9. k)x22x8=0k) \: x^2 - 2x - 8 = 0

    • Solutions: x=2x = -2, x=4x = 4
  10. l)6x2=10x+14l) \: 6x^2 = 10x + 14

    • Solutions: x=561096x = \frac{5}{6} - \frac{\sqrt{109}}{6}, x=56+1096x = \frac{5}{6} + \frac{\sqrt{109}}{6}
  11. m)7x224=2(3x12)m) \: 7x^2 - 24 = 2(3x - 12)

    • Solutions: x=0x = 0, x=67x = \frac{6}{7}
  12. n)x2+5x28=0n) \: x^2 + 5x - 28 = 0

    • Solutions: x=5+1372x = \frac{-5 + \sqrt{137}}{2}, x=51372x = \frac{-5 - \sqrt{137}}{2}

Would you like a detailed explanation for any of these solutions?

Here are 5 related questions to explore:

  1. How do you find the discriminant of a quadratic equation?
  2. What is the quadratic formula and when is it used?
  3. Can you explain how to complete the square for solving quadratics?
  4. How do imaginary solutions arise in quadratic equations?
  5. How do you factor quadratic equations?

Tip: When solving quadratic equations, always check if you can simplify or factor the equation first before using the quadratic formula.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations

Formulas

Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Theorems

Quadratic Formula
Discriminant

Suitable Grade Level

Grades 9-11