Solve x^2-16x=-54

$${ x }^{ 2 } -16x=-54$$

Solve for x

$$x=\sqrt{10}+8\approx 11.16227766$$

$$x=8-\sqrt{10}\approx 4.83772234$$

Steps Using the Quadratic Formula

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.

$$x^{2}-16x=-54$$

Add 54 to both sides of the equation.

$$x^{2}-16x-\left(-54\right)=-54-\left(-54\right)$$

Subtracting -54 from itself leaves 0.

$$x^{2}-16x-\left(-54\right)=0$$

Subtract -54 from 0.

$$x^{2}-16x+54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 54}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 54}}{2}$$

Multiply -4 times 54.

$$x=\frac{-\left(-16\right)±\sqrt{256-216}}{2}$$

Add 256 to -216.

$$x=\frac{-\left(-16\right)±\sqrt{40}}{2}$$

Take the square root of 40.

$$x=\frac{-\left(-16\right)±2\sqrt{10}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{10}}{2}$$

Now solve the equation x=\frac{16±2\sqrt{10}}{2} when ± is plus. Add 16 to 2\sqrt{10}.

$$x=\frac{2\sqrt{10}+16}{2}$$

Divide 16+2\sqrt{10} by 2.

$$x=\sqrt{10}+8$$

Now solve the equation x=\frac{16±2\sqrt{10}}{2} when ± is minus. Subtract 2\sqrt{10} from 16.

$$x=\frac{16-2\sqrt{10}}{2}$$

Divide 16-2\sqrt{10} by 2.

$$x=8-\sqrt{10}$$

The equation is now solved.

$$x=\sqrt{10}+8 x=8-\sqrt{10}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-16x=-54$$

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.

$$x^{2}-16x+\left(-8\right)^{2}=-54+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-54+64$$

Add -54 to 64.

$$x^{2}-16x+64=10$$

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}.

$$\left(x-8\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{10}$$

Simplify.

$$x-8=\sqrt{10} x-8=-\sqrt{10}$$

Add 8 to both sides of the equation.

$$x=\sqrt{10}+8 x=8-\sqrt{10}$$