Solve (x^2-2)^2=x+2

$$( x ^ { 2 } - 2 ) ^ { 2 } = x + 2$$

Solve for x

$$x=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

$$x=2$$

$$x=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

$$x=-1$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-2\right)^{2}.

$$\left(x^{2}\right)^{2}-4x^{2}+4=x+2$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-4x^{2}+4=x+2$$

Subtract x from both sides.

$$x^{4}-4x^{2}+4-x=2$$

Subtract 2 from both sides.

$$x^{4}-4x^{2}+4-x-2=0$$

Subtract 2 from 4 to get 2.

$$x^{4}-4x^{2}+2-x=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{4}-4x^{2}-x+2=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-4x^{2}-x+2 by x+1 to get x^{3}-x^{2}-3x+2. Solve the equation where the result equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-x^{2}-3x+2 by x-2 to get x^{2}+x-1. Solve the equation where the result equals to 0.

$$x^{2}+x-1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 1 for b, and -1 for c in the quadratic formula.

$$x=\frac{-1±\sqrt{1^{2}-4\times 1\left(-1\right)}}{2}$$

Do the calculations.

$$x=\frac{-1±\sqrt{5}}{2}$$

Solve the equation x^{2}+x-1=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$

List all found solutions.

$$x=-1 x=2 x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$