Transcript No Slide Title

Solving Equations That Lead to Quadratic Equations
• There are several methods one can use to solve a
quadratic equation. Sometimes we are called upon to
solve an equation that is not quadratic, but the
solution process leads to a quadratic equation. Two
examples of this will be given.
• Recall that a quadratic equation is an equation that
can be simplified into the form …
ax  bx  c  0
2
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• Example 1:
2
4

1
x2 x3
Solve the equation
Multiply every term by the
common denominator to yield
2 x  3  4( x  2)  1( x  2)( x  3)
Simplify
2x  6  4x  8  x  x  6
2
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2x  6  4x  8  x  x  6
2
Note that this is a quadratic equation. Further
simplifying leads to …
6x  2  x  x  6
2
0  x  5x  4
2
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x2  5x  4  0
Use the quadratic formula to complete the problem.
5  25  4(1)(4)
x
2(1)
5  41
x
2
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5  41
x
2
Note that neither of these solutions makes a
denominator in the original rational equation zero,
so both solutions are valid.
2
4

1
x2 x3
Note also that we started with a rational equation, but
the solution process lead to a quadratic equation.
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• Example 2:
2  x  2x  3
Solve the equation
Square both sides …
x  4x  4  2x  3
2
Simplify
x  2x 1  0
2
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x  2x 1  0
2
Factor and solve.
 x  1
2
0
x 1  0
x  1
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x  1
Note that this solution checks in the original
equation …
2   1  2  1  3
1 1
11
Here we started with a radical equation, but the
solution process lead to a quadratic equation.
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