A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are known as the roots of the equation and can be found using various methods, including factoring, completing the square, or using the quadratic formula.
Quadratic Formula
The quadratic formula is a standard method for solving any quadratic equation. It is given by: x = (-b ± √(b² – 4ac)) / 2a. This formula calculates the roots by considering the coefficients of the quadratic equation.
Example 1: Solving Using the Quadratic Formula
Solve the quadratic equation 2x² + 3x – 2 = 0.
Using the quadratic formula, x = (-3 ± √(3² – 4 * 2 * -2)) / (2 * 2).
Calculation: x = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4 = (-3 ± 5) / 4.
The solutions are x = (-3 + 5) / 4 = 1/2, and x = (-3 – 5) / 4 = -2.
Example 2: Solving Using Factoring
Solve the quadratic equation x² – 5x + 6 = 0.
First, factor the quadratic equation: (x – 2)(x – 3) = 0.
Setting each factor equal to zero gives x – 2 = 0 or x – 3 = 0.
So, the solutions are x = 2 and x = 3.
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