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Algebra
Question

Solving an Equation                 In Exercises                use the formula on page $$450$$ to find all solutions of the equation and represent the solutions graphically.

$$x ^ { 4 } + 81 = 0$$

$$x ^ { 4 } + 81 = 0$$

Subtract $$81$$ from both sides

$$x ^ { 4 } + 81 - 81 = 0 - 81$$

Simplify

$$x ^ { 4 } = - 81$$

Rewrite the equation with $$u = x ^ { 2 }$$ and $$u ^ { 2 } = x ^ { 4 }$$

$$u ^ { 2 } = - 81$$

Solve $$u ^ { 2 } = - 81 : u = 9 i , u = - 9 i$$

$$u = 9 i , u = - 9 i$$

Substitute back $$u = x ^ { 2 } ,$$ solve for $$x$$

Solve$$x ^ { 2 } = 9 i : x = \frac { 3 \sqrt { 2 } } { 2 } + \frac { 3 \sqrt { 2 } } { 2 } i , x = - \frac { 3 \sqrt { 2 } } { 2 } - \frac { 3 \sqrt { 2 } } { 2 } i$$

Solve $$x ^ { 2 } = - 9 i : x = - \frac { 3 \sqrt { 2 } } { 2 } + \frac { 3 \sqrt { 2 } } { 2 } i , x = \frac { 3 \sqrt { 2 } } { 2 } - \frac { 3 \sqrt { 2 } } { 2 } i$$

The solutions are

$$x = \frac { 3 \sqrt { 2 } } { 2 } + \frac { 3 \sqrt { 2 } } { 2 } i , x = - \frac { 3 \sqrt { 2 } } { 2 } - \frac { 3 \sqrt { 2 } } { 2 } i , x = - \frac { 3 \sqrt { 2 } } { 2 } + \frac { 3 \sqrt { 2 } } { 2 } i , x = \frac { 3 \sqrt { 2 } } { 2 } - \frac { 3 \sqrt { 2 } } { 2 } i$$

Solution
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