Solve x^2-5x+6=x

$$x ^ { 2 } - 5 x + 6 = x$$

Solve for x

$$x=\sqrt{3}+3\approx 4.732050808$$

$$x=3-\sqrt{3}\approx 1.267949192$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$x^{2}-5x+6-x=0$$

Combine -5x and -x to get -6x.

$$x^{2}-6x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-\left(-6\right)±\sqrt{36-24}}{2}$$

Add 36 to -24.

$$x=\frac{-\left(-6\right)±\sqrt{12}}{2}$$

Take the square root of 12.

$$x=\frac{-\left(-6\right)±2\sqrt{3}}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{3}}{2}$$

Now solve the equation x=\frac{6±2\sqrt{3}}{2} when ± is plus. Add 6 to 2\sqrt{3}.

$$x=\frac{2\sqrt{3}+6}{2}$$

Divide 6+2\sqrt{3} by 2.

$$x=\sqrt{3}+3$$

Now solve the equation x=\frac{6±2\sqrt{3}}{2} when ± is minus. Subtract 2\sqrt{3} from 6.

$$x=\frac{6-2\sqrt{3}}{2}$$

Divide 6-2\sqrt{3} by 2.

$$x=3-\sqrt{3}$$

The equation is now solved.

$$x=\sqrt{3}+3 x=3-\sqrt{3}$$

Steps for Completing the Square

Subtract x from both sides.

$$x^{2}-5x+6-x=0$$

Combine -5x and -x to get -6x.

$$x^{2}-6x+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-6x=-6$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-6+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-6+9$$

Add -6 to 9.

$$x^{2}-6x+9=3$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-3=\sqrt{3} x-3=-\sqrt{3}$$

Add 3 to both sides of the equation.

$$x=\sqrt{3}+3 x=3-\sqrt{3}$$