Solving Quadratic Equations

Examples of Solving Quadratic Equations

Example 1:

Solve $x^2 - 3x - 4 = 0$

Solution: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4*1*(-4)}}{2*1} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2}$, so $x = -1, 4$

Example 2:

Solve $x^2 + 2x - 8 = 0$

Solution: $x = \frac{-2 \pm \sqrt{2^2 - 4*1*(-8)}}{2*1} = \frac{-2 \pm \sqrt{4 + 32}}{2} = \frac{-2 \pm 6}{2}$, so $x = -4, 2$

Example 3:

Solve $2x^2 - 5x + 3 = 0$

Solution: $x = \frac{5 \pm \sqrt{(-5)^2 - 4*2*3}}{2*2} = \frac{5 \pm \sqrt{25 - 24}}{4} = \frac{5 \pm 1}{4}$, so $x = 1, \frac{3}{2}$

Example 4:

Solve $x^2 - 4 = 0$

Solution: $x = \frac{-0 \pm \sqrt{0 - 4*1*(-4)}}{2*1} = \frac{0 \pm \sqrt{16}}{2} = \frac{0 \pm 4}{2}$, so $x = -2, 2$

Example 5:

Solve $3x^2 - 6x + 3 = 0$

Solution: $x = \frac{6 \pm \sqrt{(-6)^2 - 4*3*3}}{2*3} = \frac{6 \pm \sqrt{36 - 36}}{6} = 1$