Solve x^2+16x+64=15

$$x ^ { 2 } + 16 x + 64 = 15$$

Solve for x (complex solution)

$$x=\sqrt{15}-8\approx -4.127016654$$

$$x=-\left(\sqrt{15}+8\right)\approx -11.872983346$$

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.

$$x^{2}+16x+64=15$$

Subtract 15 from both sides of the equation.

$$x^{2}+16x+64-15=15-15$$

Subtracting 15 from itself leaves 0.

$$x^{2}+16x+64-15=0$$

Subtract 15 from 64.

$$x^{2}+16x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 49}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 49}}{2}$$

Multiply -4 times 49.

$$x=\frac{-16±\sqrt{256-196}}{2}$$

Add 256 to -196.

$$x=\frac{-16±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-16±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{-16±2\sqrt{15}}{2} when ± is plus. Add -16 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-16}{2}$$

Divide -16+2\sqrt{15} by 2.

$$x=\sqrt{15}-8$$

Now solve the equation x=\frac{-16±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from -16.

$$x=\frac{-2\sqrt{15}-16}{2}$$

Divide -16-2\sqrt{15} by 2.

$$x=-\sqrt{15}-8$$

The equation is now solved.

$$x=\sqrt{15}-8 x=-\sqrt{15}-8$$

Steps for Completing the Square

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}.

$$\left(x+8\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{15}$$

Simplify.

$$x+8=\sqrt{15} x+8=-\sqrt{15}$$

Subtract 8 from both sides of the equation.

$$x=\sqrt{15}-8 x=-\sqrt{15}-8$$

Solve for x

$$x=\sqrt{15}-8\approx -4.127016654$$

$$x=-\sqrt{15}-8\approx -11.872983346$$

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.

$$x^{2}+16x+64=15$$

Subtract 15 from both sides of the equation.

$$x^{2}+16x+64-15=15-15$$

Subtracting 15 from itself leaves 0.

$$x^{2}+16x+64-15=0$$

Subtract 15 from 64.

$$x^{2}+16x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 49}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 49}}{2}$$

Multiply -4 times 49.

$$x=\frac{-16±\sqrt{256-196}}{2}$$

Add 256 to -196.

$$x=\frac{-16±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-16±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{-16±2\sqrt{15}}{2} when ± is plus. Add -16 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-16}{2}$$

Divide -16+2\sqrt{15} by 2.

$$x=\sqrt{15}-8$$

Now solve the equation x=\frac{-16±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from -16.

$$x=\frac{-2\sqrt{15}-16}{2}$$

Divide -16-2\sqrt{15} by 2.

$$x=-\sqrt{15}-8$$

The equation is now solved.

$$x=\sqrt{15}-8 x=-\sqrt{15}-8$$

Steps for Completing the Square

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}.

$$\left(x+8\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{15}$$

Simplify.

$$x+8=\sqrt{15} x+8=-\sqrt{15}$$

Subtract 8 from both sides of the equation.

$$x=\sqrt{15}-8 x=-\sqrt{15}-8$$