Solving Quadratic Equations Using the Quadratic Formula

Quadratic Equations

Solving Quadratic Equations Using the Quadratic Formula

Quadratic equations are polynomial equations of degree two, typically written in the standard form ax2 + bx + c = 0. The quadratic formula provides a systematic method for finding the solutions (roots) of any quadratic equation.

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

• Discriminant: The expression under the square root, $b^2 - 4ac$, is called the discriminant. It determines the nature of the roots:

  • If $b^2 - 4ac > 0$, there are two distinct real solutions.

  • If $b^2 - 4ac = 0$, there is one real solution (a repeated root).

  • If $b^2 - 4ac < 0$, there are two complex solutions.

Example: Solve $x^2 - 10x + 41 = 0$

• Identify coefficients: $a = 1$, $b = -10$, $c = 41$

• Apply the quadratic formula: $x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$ $x = \frac{10 \pm \sqrt{100 - 164}}{2}$ $x = \frac{10 \pm \sqrt{-64}}{2}$ $x = \frac{10 \pm 8i}{2}$ $x = 5 \pm 4i$

• Solution Set: $\{5 + 4i,\ 5 - 4i\}$

Key Points:

• Always write the quadratic equation in standard form before applying the formula.

• Simplify the radical and the final answer as much as possible.

• Use i to represent the imaginary unit when the discriminant is negative.