The Quadratic Formula

Introduction to the Formula

The Quadratic Formula is a powerful tool in algebra which provides a method to solve any quadratic equation, regardless of the coefficients.
Given a quadratic equation in the form ax² + bx + c = 0, the Quadratic Formula is: x = [-b ± sqrt(b² - 4ac)] / (2a).
Note on symbols: “x” represents the unknown, “±” means plus or minus, “sqrt” indicates square root, and “a”, “b”, “c” are the coefficients from the equation.

The symbol “±” indicates that there are usually two solutions to a quadratic - one when you use ”+” (plus), and another when you use ”-“ (minus).
The expression underneath the square root sign, (b² - 4ac), is called the discriminant and can indicate the nature of the solutions.
If the discriminant is positive, there will be two real and distinct solutions.
If the discriminant equals zero, there will be one real and repeated solution.
If the discriminant is negative, there will be two complex solutions (don’t worry, these don’t typically appear in algebra at this level!).

To use the Quadratic Formula, first write your equation in the form ax² + bx + c = 0 and identify the coefficients a, b and c.
Then substitute these values in place of a, b and c in the formula: x = [-b ± sqrt(b² - 4ac)] / (2a).
Next, simplify the expression under the square root (the discriminant), calculate the ± (plus or minus) part, and then divide by 2a.

Let’s use the Quadratic Formula to solve the equation 2x² - 3x - 2 = 0.
Here a = 2, b = -3, c = -2.
Substituting these into the formula gives: x = [3 ± sqrt((-3)² - 42(-2)) ] / (2*2).
Simplifying this gives: x = [3 ± sqrt(9+16)] / 4 => x = 1 or x = -2.

Study and practice the Quadratic Formula; it’s invaluable for solving quadratic equations quickly and accurately.