Solve x^2+(x+1)^2=1

$$x ^ { 2 } + ( x + 1 ) ^ { 2 } = 1$$

Solve for x

$$x=-1$$

$$x=0$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

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

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}}}{2\times 2}$$

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±2}{4}$$

Now solve the equation x=\frac{-2±2}{4} when ± is plus. Add -2 to 2.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

Now solve the equation x=\frac{-2±2}{4} when ± is minus. Subtract 2 from -2.

$$x=-\frac{4}{4}$$

Divide -4 by 4.

$$x=-1$$

The equation is now solved.

$$x=0 x=-1$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

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

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

Divide both sides by 2.

$$\frac{2x^{2}+2x}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{2}{2}x=\frac{0}{2}$$

Divide 2 by 2.

$$x^{2}+x=\frac{0}{2}$$

Divide 0 by 2.

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

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{1}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

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

Simplify.

$$x+\frac{1}{2}=\frac{1}{2} x+\frac{1}{2}=-\frac{1}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=0 x=-1$$