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The Quadratic Formula

The Quadratic Formula. 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:.

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The Quadratic Formula

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  1. The Quadratic Formula 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

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

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

  4. If the quadratic expression is in the form x2+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.

  5. Deriving the Quadratic Formula • You’ll solve 2x2+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, ax2+bx+c=0, for the equation 2x2+3x-1=0. • Group all the variable terms on the left side of your equation so that it is in the form ax2+bx=-c.

  6. It’s easiest to complete the square when the coefficient of x2 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

  7. 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 • Take the square root of both sides of your equation, like this:

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

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

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

  11. Example A • Use the quadratic formula to solve 3x2+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 andor about 0.907 and -2.573.

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