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Other Types of Equations
Equations with Radical and Rational Exponents
When solving an equation linear in form with radicals or
rational exponents , simplify the equation, isolate the most
complicated radical on one side , and raise both sides to
the power equal to the index of the radical in order to
eliminate the last . You may need to repeat this procedure
if the resulting equation still contains a radical.
Note: We solve radical equations over the real numbers .
Caution! If you raise both sides of an equation to
an
even power , the new equation may have more real
solutions than the original one.
Example: Equation x = 6 has solution set: {6}.
Raising both sides to the power 2, gives the equation
x^2 = 36 which has solution set x = ±6.
Thus, x = −6 is an extraneous solution to the original
equation and must be rejected.
Important! When raising to an even power, always
check each proposed solution in the original equation.
Example: Solve
Example: Solve
Note: If n is an even number, is never negative .
Example: Solve the equation
Example: Solve the equation
Equations Quadratic in Form
Example: Solve by using a substitution.
Equations Quadratic in x^2 (Biquadratic)
Example:
Solve the equation in the complex number system
Example: Find all solutions of the equation:
Example: Solve by factoring
Equations with Absolute Value
Recall:
1.x is the distance on the number line from 0 to x.
2.x≥ 0
3.x= 0 if and only if x = 0
4.x= −x
5. The algebraic definition :
If a is a positive real number and u is any algebraic
expression , then
u= a is equivalent to u = a or u = −a .
Example: Solve the equations.
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