Equations Quadratic in Form
The next example involves a fractional exponent. To identify this type of
equation as quadratic in form, recall how to square an expression with a fractional
exponent. For example, (x1/2)2 = x, (x1/4)2
= x1/2, and (x1/3)2 = x2/3.
Example
A fractional exponent
Solve x - 9x1/2 + 14 = 0.
Solution
Note that the square of x1/2 is x. Let w = x1/2; then w2
= (x1/2)2 = x. Now substitute
w and w2 into the original equation:
| |
|
w2 - 9w + 14 |
= 0 |
|
| |
|
(w - 7)(w - 2) |
= 0 |
|
| w - 7 |
= 0 |
or |
w - 2 |
= 0 |
|
| w |
= 7 |
or |
w |
= 2 |
|
| x1/2 |
= 7 |
or |
x1/2 |
= 2 |
Replace w by x1/2. |
| x |
= 49 |
or |
x |
= 4 |
Square each side. |
Because we squared each side, we must check for extraneous roots. First evaluate x
- 9x1/2 + 14 for x = 49:
49 - 9 · 491/2 + 14 = 49 - 9 · 7 + 14 = 0
Now evaluate x - 9x1/2 + 14 for x = 4:
4 - 9 · 41/2 + 14 = 4 - 9 · 2 + 14 = 0
Because each solution checks, the solution set is {4, 49}.
Caution
An equation of quadratic form must have a term that is the
square of another. Equations such as x4 - 5x3 + 6 = 0 or x1/2
- 3x1/3 - 18 = 0
are not quadratic in form and cannot be solved by substitution.
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