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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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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 Â· 491/2 + 14 = 49 - 9 Â· 7 + 14 = 0

Now evaluate x - 9x1/2 + 14 for x = 4:

4 - 9 Â· 41/2 + 14 = 4 - 9 Â· 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.