Find out the answer to "Find all the real roots ofx2+x2(x+1)2=3"

Dexter Odom

Answered question

2022-04-04

Find all the real roots of

x^{2} + \frac{x^{2}}{{(x + 1)}^{2}} = 3

pypberissootcu

Beginner2022-04-05Added 14 answers

Step 1

x^{2} + \frac{x^{2}}{{(x + 1)}^{2}} = 3

(x \neq 1)

\Rightarrow x^{2} + \frac{x^{2}}{{(x + 1)}^{2}} - 3 = 0

Which simplifies as

\frac{x^{4} + 2 x^{3} - x^{2} - 6 x - 3}{{(x + 1)}^{2}} = 0

\Rightarrow x^{4} + 2 x^{3} - x^{2} - 6 x - 3 = 0

Factorising, we get

(x^{2} - x - 1) (x^{2} + 3 x + 3) = 0

For which we get two equations to solve

x^{2} - x - 1 = 0

x^{2} + 3 x + 3 = 0

Solving the first eqn. we get

x = \frac{1 \pm \sqrt{5}}{2}

The second equation has no (real) solution.

∴ x = \frac{1 \pm \sqrt{5}}{2}

Mikaela Winters

Beginner2022-04-06Added 14 answers

Step 1
Factorize the equation as follows

0 = x^{2} + \frac{x^{2}}{{(x + 1)}^{2}} - 3

= \frac{1}{{(1 + x)}^{2}} (x^{2} {(x + 1)}^{2} + x^{2} - 3 {(x + 1)}^{2})

= \frac{1}{{(1 + x)}^{2}} (x^{4} + 2 (x + 1) x^{2} - 3 {(x + 1)}^{2})

= \frac{1}{{(1 + x)}^{2}} (x^{2} - (x + 1)) (x^{2} + 3 (x + 1))

which, from the factor

x^{2} - x - 1

, yields the real solution

x = \frac{1}{2} \pm \frac{\sqrt{5}}{2}

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