Solve x^2+10=10x

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

Solve for x

$$x=\sqrt{15}+5\approx 8.872983346$$

$$x=5-\sqrt{15}\approx 1.127016654$$

Steps Using the Quadratic Formula

Subtract 10x from both sides.

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

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.

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

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

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

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-\left(-10\right)±\sqrt{100-40}}{2}$$

Add 100 to -40.

$$x=\frac{-\left(-10\right)±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-\left(-10\right)±2\sqrt{15}}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{10±2\sqrt{15}}{2} when ± is plus. Add 10 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}+10}{2}$$

Divide 10+2\sqrt{15} by 2.

$$x=\sqrt{15}+5$$

Now solve the equation x=\frac{10±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from 10.

$$x=\frac{10-2\sqrt{15}}{2}$$

Divide 10-2\sqrt{15} by 2.

$$x=5-\sqrt{15}$$

The equation is now solved.

$$x=\sqrt{15}+5 x=5-\sqrt{15}$$

Steps for Completing the Square

Subtract 10x from both sides.

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

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

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

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-10+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-10+25$$

Add -10 to 25.

$$x^{2}-10x+25=15$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{15}$$

Simplify.

$$x-5=\sqrt{15} x-5=-\sqrt{15}$$

Add 5 to both sides of the equation.

$$x=\sqrt{15}+5 x=5-\sqrt{15}$$