Solve 2x^2+4x-480=8

$$2 x ^ { 2 } + 4 x - 480 = 8$$

Solve for x

$$x=7\sqrt{5}-1\approx 14.652475842$$

$$x=-7\sqrt{5}-1\approx -16.652475842$$

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.

$$2x^{2}+4x-480=8$$

Subtract 8 from both sides of the equation.

$$2x^{2}+4x-480-8=8-8$$

Subtracting 8 from itself leaves 0.

$$2x^{2}+4x-480-8=0$$

Subtract 8 from -480.

$$2x^{2}+4x-488=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -488 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-488\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-488\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-488\right)}}{2\times 2}$$

Multiply -8 times -488.

$$x=\frac{-4±\sqrt{16+3904}}{2\times 2}$$

Add 16 to 3904.

$$x=\frac{-4±\sqrt{3920}}{2\times 2}$$

Take the square root of 3920.

$$x=\frac{-4±28\sqrt{5}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±28\sqrt{5}}{4}$$

Now solve the equation x=\frac{-4±28\sqrt{5}}{4} when ± is plus. Add -4 to 28\sqrt{5}.

$$x=\frac{28\sqrt{5}-4}{4}$$

Divide -4+28\sqrt{5} by 4.

$$x=7\sqrt{5}-1$$

Now solve the equation x=\frac{-4±28\sqrt{5}}{4} when ± is minus. Subtract 28\sqrt{5} from -4.

$$x=\frac{-28\sqrt{5}-4}{4}$$

Divide -4-28\sqrt{5} by 4.

$$x=-7\sqrt{5}-1$$

The equation is now solved.

$$x=7\sqrt{5}-1 x=-7\sqrt{5}-1$$

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.

$$2x^{2}+4x-480=8$$

Add 480 to both sides of the equation.

$$2x^{2}+4x-480-\left(-480\right)=8-\left(-480\right)$$

Subtracting -480 from itself leaves 0.

$$2x^{2}+4x=8-\left(-480\right)$$

Subtract -480 from 8.

$$2x^{2}+4x=488$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{488}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{488}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{488}{2}$$

Divide 488 by 2.

$$x^{2}+2x=244$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=244+1^{2}$$

Square 1.

$$x^{2}+2x+1=244+1$$

Add 244 to 1.

$$x^{2}+2x+1=245$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=245$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{245}$$

Simplify.

$$x+1=7\sqrt{5} x+1=-7\sqrt{5}$$

Subtract 1 from both sides of the equation.

$$x=7\sqrt{5}-1 x=-7\sqrt{5}-1$$