11
11.2 The Quadratic Formula
Solving Quadratic Equations Using the Quadratic Formula.
By solving the general quadratic equation [pic] using the method of completing the square, one can derive the quadratic formula. The quadratic formula can be used to solve any quadratic equation.
The Quadratic Formula
The solutions of a quadratic equation in standard form [pic], with a(0, are given by the quadratic formula
[pic].
Example 1: Solve the given quadratic equations by using the quadratic formula.
[pic]
[pic]
The Discriminant
The quantity [pic], which appears under the radical sign in the quadratic formula, is called the discriminant. The value of the discriminant for a given quadratic equation can be used to determine the kinds of solutions that the quadratic equation has.
The Discriminant and the Kinds of Solutions to [pic]
|Value of the Discriminant |Kinds of Solutions |Graph of y=[pic] |
|[pic]>0 |Two unequal real solutions. | |
| |Graph crosses the x-axis twice. |[pic] |
|[pic]=0 |One real solution (a repeated solution)|[pic] |
| |that is a real number. | |
| |Graph touches the x-axis. | |
|[pic] ................
................
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