12.2. Solving a Quadratic by Formula
We have learned several calculator methods for solving a quadratic, and here we present a manual method, the quadratic formula. It will work for any quadratic, regardless of the type of roots, can be used for literal quadratic equations, and can easily be programmed for the computer. To use the formula, we must first put a quadratic equation into general form.
12.2.1. General Form of a Quadratic
A quadratic is in general form when it is written in the following form, where a, b, and c are constants:
Example 8:Write the quadratic equation
in general form, and identify a, b, and c. Solution: Subtracting 5x2/3 from both sides and writing the terms in descending order of the exponents, we obtain
Quadratics in general form are usually written without fractions and with the first term positive. Multiplying by −3, we get
The equation is now in general form, with z = 5, b = 12, and c = −21. |
12.2.2. The Quadratic Formula
We can find the roots of any quadratic equation ax2 + bx + c = 0 by using the well-known quadratic formula
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