Solve x^2+10=6x

$${ x }^{ 2 } +10=6x$$

Solve for x (complex solution)

$$x=3+i$$

$$x=3-i$$

Steps Using the Quadratic Formula

Subtract 6x from both sides.

$$x^{2}+10-6x=0$$

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.

$$x^{2}-6x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 10}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-\left(-6\right)±\sqrt{36-40}}{2}$$

Add 36 to -40.

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

Take the square root of -4.

$$x=\frac{-\left(-6\right)±2i}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2i}{2}$$

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

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

Divide 6+2i by 2.

$$x=3+i$$

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

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

Divide 6-2i by 2.

$$x=3-i$$

The equation is now solved.

$$x=3+i x=3-i$$

Steps for Completing the Square

Subtract 6x from both sides.

$$x^{2}+10-6x=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-6x=-10$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-10+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-10+9$$

Add -10 to 9.

$$x^{2}-6x+9=-1$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-3=i x-3=-i$$

Add 3 to both sides of the equation.

$$x=3+i x=3-i$$