Section%204.5%20%20%20Solving%20Exponential%20Equation

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Section 11.5
Equations Reducible to Quadratic
Phong Chau
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Some equations are not quadratic, but can be
turned into quadratic equation by using
substitution. Such equations are called Equations
in Quadratic Form or Reducible to Quadratic.
x4 5x2 4 0
(x2)2 5(x2) 4 0
u2 5u 4 0.
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Example
Solve x4 5x2 4 0.
Solution
Let u x2. Then we solve by substituting u for
x2 and u2 for x4
u2 5u 4 0
(u 1)(u 4) 0
Factoring
Principle of zero products
u 1 0 or u 4 0
u 1 or u 4
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x2 1 or x2 4
Replace u with x2
Check
x 1
x 2
x4 5x2 4 0
x4 5x2 4 0
(16) 5(4) 4 0
(1) 5(1) 4 0
TRUE
TRUE
The solutions are 1, 1, 2, and 2.
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Example
Solve
Solution
u2 8u 9 0
(u 9)(u 1) 0
u 9 0 or u 1 0
u 9 or u 1
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Check
x 81
x 1
FALSE
TRUE
The solution is 81.
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Example
Solve
Solution
Let u t -1. Then we solve by substituting u
for t -1 and u2 for t -2
u2 4u 2 0
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Examples
Solve the following equations