Solve x^2-11x+56=5x+13

$${ x }^{ 2 } -11x+56=5x+13$$

Solve for x

$$x=\sqrt{21}+8\approx 12.582575695$$

$$x=8-\sqrt{21}\approx 3.417424305$$

Steps Using the Quadratic Formula

Subtract 5x from both sides.

$$x^{2}-11x+56-5x=13$$

Combine -11x and -5x to get -16x.

$$x^{2}-16x+56=13$$

Subtract 13 from both sides.

$$x^{2}-16x+56-13=0$$

Subtract 13 from 56 to get 43.

$$x^{2}-16x+43=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 43 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 43}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 43}}{2}$$

Multiply -4 times 43.

$$x=\frac{-\left(-16\right)±\sqrt{256-172}}{2}$$

Add 256 to -172.

$$x=\frac{-\left(-16\right)±\sqrt{84}}{2}$$

Take the square root of 84.

$$x=\frac{-\left(-16\right)±2\sqrt{21}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{21}}{2}$$

Now solve the equation x=\frac{16±2\sqrt{21}}{2} when ± is plus. Add 16 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}+16}{2}$$

Divide 16+2\sqrt{21} by 2.

$$x=\sqrt{21}+8$$

Now solve the equation x=\frac{16±2\sqrt{21}}{2} when ± is minus. Subtract 2\sqrt{21} from 16.

$$x=\frac{16-2\sqrt{21}}{2}$$

Divide 16-2\sqrt{21} by 2.

$$x=8-\sqrt{21}$$

The equation is now solved.

$$x=\sqrt{21}+8 x=8-\sqrt{21}$$

Steps for Completing the Square

Subtract 5x from both sides.

$$x^{2}-11x+56-5x=13$$

Combine -11x and -5x to get -16x.

$$x^{2}-16x+56=13$$

Subtract 56 from both sides.

$$x^{2}-16x=13-56$$

Subtract 56 from 13 to get -43.

$$x^{2}-16x=-43$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-43+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-43+64$$

Add -43 to 64.

$$x^{2}-16x+64=21$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=21$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{21}$$

Simplify.

$$x-8=\sqrt{21} x-8=-\sqrt{21}$$

Add 8 to both sides of the equation.

$$x=\sqrt{21}+8 x=8-\sqrt{21}$$