Solve x^2+8x=2400

$${ x }^{ 2 } +8x = 2400$$

Solve for x

$$x=4\sqrt{151}-4\approx 45.15282291$$

$$x=-4\sqrt{151}-4\approx -53.15282291$$

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.

$$x^{2}+8x=2400$$

Subtract 2400 from both sides of the equation.

$$x^{2}+8x-2400=2400-2400$$

Subtracting 2400 from itself leaves 0.

$$x^{2}+8x-2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2400\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-2400\right)}}{2}$$

Multiply -4 times -2400.

$$x=\frac{-8±\sqrt{64+9600}}{2}$$

Add 64 to 9600.

$$x=\frac{-8±\sqrt{9664}}{2}$$

Take the square root of 9664.

$$x=\frac{-8±8\sqrt{151}}{2}$$

Now solve the equation x=\frac{-8±8\sqrt{151}}{2} when ± is plus. Add -8 to 8\sqrt{151}.

$$x=\frac{8\sqrt{151}-8}{2}$$

Divide -8+8\sqrt{151} by 2.

$$x=4\sqrt{151}-4$$

Now solve the equation x=\frac{-8±8\sqrt{151}}{2} when ± is minus. Subtract 8\sqrt{151} from -8.

$$x=\frac{-8\sqrt{151}-8}{2}$$

Divide -8-8\sqrt{151} by 2.

$$x=-4\sqrt{151}-4$$

The equation is now solved.

$$x=4\sqrt{151}-4 x=-4\sqrt{151}-4$$

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.

$$x^{2}+8x=2400$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=2400+4^{2}$$

Square 4.

$$x^{2}+8x+16=2400+16$$

Add 2400 to 16.

$$x^{2}+8x+16=2416$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=2416$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{2416}$$

Simplify.

$$x+4=4\sqrt{151} x+4=-4\sqrt{151}$$

Subtract 4 from both sides of the equation.

$$x=4\sqrt{151}-4 x=-4\sqrt{151}-4$$