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# Solving Quadratic Equations by Using the Quadratic Formula

Objective Introduce the Quadratic Formula and learn how to use it to solve quadratic equations.

In this lesson, you will use the Quadratic Formula to solve quadratic equations. The goal is that you become comfortable using this formula. This will require lots of practice, and so you should pay special attention to the examples.

The Quadratic Formula The solutions of a quadratic equation in the form ax 2 + bx + c = 0, where a 0, are given by the formula .

This formula can be used to find the solutions to any quadratic equation.

Notice the sign in the formula. This means that there may be two solutions to a quadratic equation. Recall that a parabola intersects the x -axis in two places, the corresponding equation has two solutions. This corresponds to the two solutions .

The expression inside of the square root, b 2 - 4ac, is called the discriminant. When the discriminant is positive, there are two distinct solutions. When this expression is zero, the square root in the Quadratic Formula is zero, and then there is only one solution, . This corresponds to the parabola intersecting the x -axis in only one point, namely the vertex. The final case is when the discriminant b 2 - 4ac is negative, in which case the square root is not defined. In this case, there are no real solutions to the quadratic equation. This corresponds to the parabola not intersecting the x-axis at all.

 Solutions of the Quadratic Equation ax 2 + bx + c = 0 Discriminant b 2 - 4ac > 0 b 2 - 4ac = 0 b 2 - 4ac < 0 Number of Solutions 2 1 0 Example Parabola Intersects the x-axis yes, in two distinct points yes, in exactly one point, the vertex no

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