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The Quadratic Formula

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The Quadratic Formula

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The Quadratic Formula

The Quadratic Formula

The Quadratic Formula

The Quadratic Formula

The Quadratic Formula

Trinomials are not always easy to factor. In fact, some trinomials cannot be factored. Thus, we need a different way to solve quadratic equations. Herein lies the importance of the quadratic formula:

Given a quadratic equation ax 2 + bx + c = 0 , the solutions are given by the equation

x =    

Example 1: Solve for x : x 2 + 8x + 15.75 = 0

a = 1 , b = 8 , and c = 15.75 .
x =

=
=
=
= or
= - or -
Thus, x = - or x = - .

Example 2: Solve for x : 3x 2 - 10x - 25 = 0 .

a = 3 , b = - 10 , and c = - 25 .
x =

=
=
=
=
= or
= 5 or -
Thus, x = 5 or x = - .

Example 3: Solve for x : -3x 2 - 24x - 48 = 0 .

a = - 3 , b = - 24 , and c = - 48 .
x =

=
=
=
=
= = - 4
Thus, x = - 4 .

Example 4: Solve for x : 2x 2 - 4x + 7 .

a = 2 , b = - 4 , and c = 7 .
x =

=
=
=
Since we cannot take the square root of a negative number, there are no solutions. (The graph of this quadratic polynomial will therefore be a parabola that never touches the x -axis.)

The Discriminant

As we have seen, there can be 0 , 1 , or 2 solutions to a quadratic equation, depending on whether the expression inside the square root sign, (b 2 - 4ac) , is positive, negative, or zero. This expression has a special name: the discriminant.

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