Solve (x+x^2)^2+(x^2-x)^2=4

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

Solve for x (complex solution)

$$x=-1$$

$$x=1$$

$$x=\sqrt{2}i\approx 1.414213562i$$

$$x=-\sqrt{2}i\approx -0-1.414213562i$$

Steps Using the Quadratic Formula

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

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$x^{2}+2x^{3}+\left(x^{2}\right)^{2}+\left(x^{2}-x\right)^{2}=4$$

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

$$x^{2}+2x^{3}+x^{4}+\left(x^{2}-x\right)^{2}=4$$

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

$$x^{2}+2x^{3}+x^{4}+\left(x^{2}\right)^{2}-2x^{2}x+x^{2}=4$$

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

$$x^{2}+2x^{3}+x^{4}+x^{4}-2x^{2}x+x^{2}=4$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{2}+2x^{3}+x^{4}+x^{4}-2x^{3}+x^{2}=4$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$x^{2}+2x^{3}+2x^{4}-2x^{3}+x^{2}=4$$

Combine 2x^{3} and -2x^{3} to get 0.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Subtract 4 from both sides.

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

Substitute t for x^{2}.

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

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 2 for a, 2 for b, and -4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-2±6}{4}$$

Solve the equation t=\frac{-2±6}{4} when ± is plus and when ± is minus.

$$t=1 t=-2$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=-1 x=1 x=-\sqrt{2}i x=\sqrt{2}i$$

Solve for x

$$x=-1$$

$$x=1$$

Steps Using the Quadratic Formula

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

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$x^{2}+2x^{3}+\left(x^{2}\right)^{2}+\left(x^{2}-x\right)^{2}=4$$

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

$$x^{2}+2x^{3}+x^{4}+\left(x^{2}-x\right)^{2}=4$$

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

$$x^{2}+2x^{3}+x^{4}+\left(x^{2}\right)^{2}-2x^{2}x+x^{2}=4$$

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

$$x^{2}+2x^{3}+x^{4}+x^{4}-2x^{2}x+x^{2}=4$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{2}+2x^{3}+x^{4}+x^{4}-2x^{3}+x^{2}=4$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$x^{2}+2x^{3}+2x^{4}-2x^{3}+x^{2}=4$$

Combine 2x^{3} and -2x^{3} to get 0.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Subtract 4 from both sides.

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

Substitute t for x^{2}.

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

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 2 for a, 2 for b, and -4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-2±6}{4}$$

Solve the equation t=\frac{-2±6}{4} when ± is plus and when ± is minus.

$$t=1 t=-2$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=1 x=-1$$