The Quadratic Formula

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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

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Recall that you can solve some quadratic
equations symbolically by recognizing their forms
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You can also undo the order of operations in
other quadratic equations when there is no
x-term, as in these
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If the quadratic expression is in the form
x2bxc, you can complete the square by using a
rectangle diagram.
In the investigation youll use the
completing-the-square method to derive the
quadratic formula.
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Deriving the Quadratic Formula

• Youll solve 2x23x-10 and develop the quadratic
  formula for the general case in the process.
• Identify the values of a, b, and c in the general
  form, ax2bxc0, for the equation 2x23x-10.
• Group all the variable terms on the left side of
  your equation so that it is in the form
  ax2bx-c.

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• Its 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

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• 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

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• Rewrite as 2a. Then get x by itself on
  the left side, like this

• There are two possible solutions given by the
  equations

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• 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?

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

• Use the quadratic formula to solve 3x25x-70.
• The equation is already in general form, so
  identify the values of a, b, and c. For this
  equation, a3, b5, and c7.

The two exact roots of the equation are
andor about 0.907 and -2.573.