Solve x^2-4x+2/-x^2+x-4=0

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

Solve for x

$$x=\sqrt{2}+2\approx 3.414213562$$

$$x=2-\sqrt{2}\approx 0.585786438$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by -x^{2}+x-4.

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

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

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

Square -4.

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

Multiply -4 times 2.

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

Add 16 to -8.

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

Take the square root of 8.

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

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{2}}{2}$$

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

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

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

$$x=\sqrt{2}+2$$

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

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

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

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

The equation is now solved.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$

Steps for Completing the Square

Multiply both sides of the equation by -x^{2}+x-4.

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

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

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Add -2 to 4.

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

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$