Solve x^2=-x^2+4x+1

$$x ^ { 2 } = - x ^ { 2 } + 4 x + 1$$

Solve for x

$$x=\frac{\sqrt{6}}{2}+1\approx 2.224744871$$

$$x=-\frac{\sqrt{6}}{2}+1\approx -0.224744871$$

Steps Using the Quadratic Formula

Add x^{2} to both sides.

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

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

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

Subtract 4x from both sides.

$$2x^{2}-4x=1$$

Subtract 1 from both sides.

$$2x^{2}-4x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-1\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-1\right)}}{2\times 2}$$

Multiply -8 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+8}}{2\times 2}$$

Add 16 to 8.

$$x=\frac{-\left(-4\right)±\sqrt{24}}{2\times 2}$$

Take the square root of 24.

$$x=\frac{-\left(-4\right)±2\sqrt{6}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{6}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±2\sqrt{6}}{4}$$

Now solve the equation x=\frac{4±2\sqrt{6}}{4} when ± is plus. Add 4 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}+4}{4}$$

Divide 4+2\sqrt{6} by 4.

$$x=\frac{\sqrt{6}}{2}+1$$

Now solve the equation x=\frac{4±2\sqrt{6}}{4} when ± is minus. Subtract 2\sqrt{6} from 4.

$$x=\frac{4-2\sqrt{6}}{4}$$

Divide 4-2\sqrt{6} by 4.

$$x=-\frac{\sqrt{6}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{6}}{2}+1 x=-\frac{\sqrt{6}}{2}+1$$

Steps for Completing the Square

Add x^{2} to both sides.

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

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

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

Subtract 4x from both sides.

$$2x^{2}-4x=1$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -4 by 2.

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

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

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

Add \frac{1}{2} to 1.

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

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

$$\left(x-1\right)^{2}=\frac{3}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{3}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{6}}{2} x-1=-\frac{\sqrt{6}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{6}}{2}+1 x=-\frac{\sqrt{6}}{2}+1$$