Solving Quadratic Equations by the Quadratic Formula

Quadratic Functions and Equations

Solving Quadratic Equations Using the Quadratic Formula

Quadratic equations are equations of the form ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The quadratic formula provides a universal method for finding the solutions (roots) of any quadratic equation.

• Quadratic Formula: The solutions to the equation ax2 + bx + c = 0 are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: The expression under the square root, b2 - 4ac, is called the discriminant. It determines the nature of the roots:

  • If b2 - 4ac > 0, there are two distinct real roots.

  • If b2 - 4ac = 0, there is one real root (a repeated root).

  • If b2 - 4ac < 0, there are two complex roots.

Example Problem

Solve the equation $x^2 - 2x - 14 = 0$ using the quadratic formula.

• Identify coefficients: $a = 1$, $b = -2$, $c = -14$

• Substitute into the quadratic formula: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-14)}}{2(1)}$ $x = \frac{2 \pm \sqrt{4 + 56}}{2}$ $x = \frac{2 \pm \sqrt{60}}{2}$ $x = \frac{2 \pm 2\sqrt{15}}{2}$ $x = 1 \pm \sqrt{15}$

• Final Answer: $x = 1 + \sqrt{15}$ and $x = 1 - \sqrt{15}$

Additional info: The quadratic formula is a fundamental tool in algebra and is widely used in trigonometry for solving equations involving trigonometric identities that reduce to quadratic form.