Solve x^2+3=6x

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

Solve for x

$$x=\sqrt{6}+3\approx 5.449489743$$

$$x=3-\sqrt{6}\approx 0.550510257$$

Steps Using the Quadratic Formula

Subtract 6x from both sides.

$$x^{2}+3-6x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-6x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12}}{2}$$

Add 36 to -12.

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

Take the square root of 24.

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

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{6}}{2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

$$x=\sqrt{6}+3 x=3-\sqrt{6}$$

Steps for Completing the Square

Subtract 6x from both sides.

$$x^{2}+3-6x=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Square -3.

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

Add -3 to 9.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 3 to both sides of the equation.

$$x=\sqrt{6}+3 x=3-\sqrt{6}$$