Quadratic Equations and Their Solutions

Quadratic Equations

• A quadratic equation has the highest power of 2.

• Example: $x^2 - 3x + 1 = 0$.

Methods of Solving Quadratic Equations

• Four methods:

  • Factorization (where possible)

  • Completing the square

  • Quadratic formula (not used in this course)

  • Graphically (see Chapter 13)

• Course focus on methods i, ii, and iv.

Solving Quadratic Equations by Factorization

Example: Solve $2x^2 - 5x - 3 = 0$

• Factorize to:
  $(2x + 1)(x - 3) = 0$

• Set each factor to zero:

  • $2x + 1 = 0 <br>x = -2$

  • $x - 3 = 0 <br>x = 3$

• Roots are $x = -2$ and $x = 3$.

Example Problem 1: Solve $x^2 + 2x - 8 = 0$

• Factors of -8: +8 and -1, -8 and +1, +4 and -2, -4 and +2.

• Only combination giving +2x is: $x^2 + 2x - 8 = (x + 4)(x - 2)$.

• Solve:

  • $x + 4 = 0 <br>x = -4$

  • $x - 2 = 0 <br>x = 2$

• Roots are $x = -4$ and $x = 2$.

Example Problem 2: Determine Roots

• (a) Solve $x^2 - 6x + 9 = 0$

  • Factor to:
    $(x - 3)(x - 3) = 0$

  • Thus, $x = 3$ is the only root.

• (b) Solve $4x^2 - 25 = 0$

  • Factor to:
    $(2x + 5)(2x - 5) = 0$

  • Roots:

  • $2x + 5 = 0 <br>x = - rac{5}{2}$

  • $2x - 5 = 0 <br>x = rac{5}{2}$

Quadratic Formula

Formula Presentation

• Form: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  • Solves quadratic equations of the form $ax^2 + bx + c = 0$.

• Example Usage:

  • Solve $x^2 - 27x + 38 = 0$ using:
    $x = \frac{27 \pm \sqrt{(-27)^2 - 4(1)(38)}}{2(1)}$.

Practical Problems Involving Quadratic Equations

Example: Shed and Path Area Problem

• Shed dimensions: 4.0m x 2.0m

• Area of path = 9.50 m²

• Path width = $t$ m.

• Area calculation:

  • Total area =
    $(4 + 2t)(2 + 2t) - 8 = 9.50$

  • Results in $4t^2 + 12t - 9.50 = 0$.

  • Solving gives:
    $t = \frac{-12 \pm \sqrt{12^2 - 4(4)(-9.50)}}{2(4)}$

• Resulting width estimate:

  • $t = 0.651 m$ (or 65 cm approx).

Simultaneous Equations

Problem: Solve Simultaneously

• Given equations:
  $y = 5x - 4 - 2x^2$
  and
  $y = 6x - 7$

• Setting them equal provides a solvable quadratic:
  $0 = -2x^2 - x + 3 <br>\implies 2x^2 + x - 3 = 0$

• For graphical solutions, plot both equations and find intersection points for solutions.

Important Note:

• Practice is essential, ensure to complete all tutorial questions to reinforce learning and understanding.