Chapter Contents

• Quadratic Equations
• 1. Solving Quadratic Equations by Factoring
• 2. Completing the Square
• 3. The Quadratic Formula
• 4. The Graph of the Quadratic Function
• 5. Equations in Quadratic Form
• Quadratic Equations Problem Solver

• Comments, Questions?

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3. The Quadratic Formula

At the end of the last section (Completing the Square), we derived a general formula for solving quadratic equations. Here is that general formula:

For any quadratic equation `ax^2+ bx + c = 0`, the solutions for x can be found by using the quadratic formula:

`x=(-b+-sqrt(b^2-4ac))/(2a)`

The expression under the square root `(b^2− 4ac = 0)`can tell us how many roots we'll get. (There's no magic here - just a consideration of what the square root of `(b^2− 4ac = 0)` is.)

If ` b^2− 4ac = 0`, then we'll have one root only, `x = −b/(2a)`.

If ` b^2− 4ac > 0`, then we'll have two roots, one involving the "+" sign and the other involving the "−" sign in the formula.

If ` b^2− 4ac < 0`, then we'll have no real roots, since you cannot find the square root of a negative number.

The expression `(b^2 − 4ac)` is called the discriminant.

Example 1

Solve 2x² - 7x - 5 = 0 using the quadratic formula.

Answer

Example 2

Solve 2x² = 4x + 3

Answer

Exercise

Solve 6r² = 6r + 1 using the quadratic formula.

Answer