We have an answer to "Solve x4 + 16 = 0 . Express..."

Selena Cowan

Answered question

2022-01-21

Solve

x^{4}

+ 16 = 0 . Express the roots in rectangular form .

KickAntitte06

Beginner2022-01-22Added 11 answers

The given expression is

x^{4}

+ 16 = 0
Adding and subtracting

8 x^{2}

, we get

x^{4} - 8 x^{2} + 8 x^{2} + 16 = 0

{(x^{2} + 4)}^{2} - 8 x^{2} = 0

{(x^{2} + 4)}^{2} - {(2 \sqrt{2 x})}^{2} = 0

(x^{2} + 4 + 2 \sqrt{2 x}) (x^{2} + 4 - 2 \sqrt{2 x}) = 0

(∵ a^{2} - b^{2} = (a + b) (a - b))

Solving the two quadratics separately.

(x^{2} + 4 + 2 \sqrt{2 x}) = 0

By quadratic formula

x = \frac{- 2 \sqrt{2 \pm \sqrt{8 - 4 \times 4}}}{2}

= \frac{- 2 \sqrt{2 \pm \sqrt{- 8}}}{2}

= \frac{- 2 \sqrt{2 \pm 2 \sqrt{2 i}}}{2}

= - \sqrt{2} \pm \sqrt{2 i}

Now , solving another quadratic equation, we get

(x^{2} + 4 - 2 \sqrt{2 x}) = 0

By quadratic formula

x = \frac{2 \sqrt{2 \pm \sqrt{8 - 4 \times 4}}}{2}

= \frac{2 \sqrt{2 \pm \sqrt{- 8}}}{2}

= \frac{2 \sqrt{2 \pm 2 \sqrt{2 i}}}{2}

= \sqrt{2} \pm \sqrt{2 i}

Hence the roots of the given expression in rectangular form are

- \sqrt{2} + \sqrt{2 i}

- \sqrt{2} - \sqrt{2 i}

\sqrt{2} + \sqrt{2 i}

\sqrt{2} - \sqrt{2 i}

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