To study the derivation of the quadratic formula To learn to use the quadratic formula To use the discriminant to determine the nature of the roots of a quadratic equation

Recall that you can solve some quadratic equations symbolically by recognizing their forms:

You can also undo the order of operations in other quadratic equations when there is no x- term, as in these:

If the quadratic expression is in the form x 2 +bx+c, you can complete the square by using a rectangle diagram. In the investigation you’ll use the completing- the-square method to derive the quadratic formula.

Deriving the Quadratic Formula You’ll solve 2x 2 +3x-1=0 and develop the quadratic formula for the general case in the process. Identify the values of a, b, and c in the general form, ax 2 +bx+c=0, for the equation 2x 2 +3x-1=0. Group all the variable terms on the left side of your equation so that it is in the form ax 2 +bx=-c.

It’s easiest to complete the square when the coefficient of x 2 is 1. So divide your equation by the value of a. Write it in the form Use a rectangle diagram to help you complete the square. What number must you add to both sides? Write your new equation in the form

o Rewrite the trinomial on the left side of your equation as a squared binomial. On the right side, find a common denominator. Write the next stage of your equation in the form o Take the square root of both sides of your equation, like this:

o Rewrite as 2a. Then get x by itself on the left side, like this: o There are two possible solutions given by the equations

o Write your two solutions in radical form. o Write your solutions in decimal form. Check them with a graph and a table. o Consider the expression What restrictions should there be so that the solutions exist and are real numbers?

If a quadratic equation is written in the general form, the roots are given by. Quadratic Formula

Example A Use the quadratic formula to solve 3x 2 +5x-7=0. The equation is already in general form, so identify the values of a, b, and c. For this equation, a=3, b=5, and c=7. The two exact roots of the equation are and or about and