Solve 360div(x+5)-360divx=1

$$360 \div (x+5)-360 \div x=1$$

Solve for x (complex solution)

$$x=\frac{-5\sqrt{287}i-5}{2}\approx -2.5-42.352685865i$$

$$x=\frac{-5+5\sqrt{287}i}{2}\approx -2.5+42.352685865i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+5\right), the least common multiple of x+5,x.

$$x\times 360-\left(x+5\right)\times 360=x\left(x+5\right)$$

Use the distributive property to multiply x+5 by 360.

$$x\times 360-\left(360x+1800\right)=x\left(x+5\right)$$

To find the opposite of 360x+1800, find the opposite of each term.

$$x\times 360-360x-1800=x\left(x+5\right)$$

Combine x\times 360 and -360x to get 0.

$$-1800=x\left(x+5\right)$$

Use the distributive property to multiply x by x+5.

$$-1800=x^{2}+5x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+5x=-1800$$

Add 1800 to both sides.

$$x^{2}+5x+1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and 1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 1800}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 1800}}{2}$$

Multiply -4 times 1800.

$$x=\frac{-5±\sqrt{25-7200}}{2}$$

Add 25 to -7200.

$$x=\frac{-5±\sqrt{-7175}}{2}$$

Take the square root of -7175.

$$x=\frac{-5±5\sqrt{287}i}{2}$$

Now solve the equation x=\frac{-5±5\sqrt{287}i}{2} when ± is plus. Add -5 to 5i\sqrt{287}.

$$x=\frac{-5+5\sqrt{287}i}{2}$$

Now solve the equation x=\frac{-5±5\sqrt{287}i}{2} when ± is minus. Subtract 5i\sqrt{287} from -5.

$$x=\frac{-5\sqrt{287}i-5}{2}$$

The equation is now solved.

$$x=\frac{-5+5\sqrt{287}i}{2} x=\frac{-5\sqrt{287}i-5}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+5\right), the least common multiple of x+5,x.

$$x\times 360-\left(x+5\right)\times 360=x\left(x+5\right)$$

Use the distributive property to multiply x+5 by 360.

$$x\times 360-\left(360x+1800\right)=x\left(x+5\right)$$

To find the opposite of 360x+1800, find the opposite of each term.

$$x\times 360-360x-1800=x\left(x+5\right)$$

Combine x\times 360 and -360x to get 0.

$$-1800=x\left(x+5\right)$$

Use the distributive property to multiply x by x+5.

$$-1800=x^{2}+5x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+5x=-1800$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-1800+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=-1800+\frac{25}{4}$$

Add -1800 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=-\frac{7175}{4}$$

Factor x^{2}+5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{2}\right)^{2}=-\frac{7175}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{-\frac{7175}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{5\sqrt{287}i}{2} x+\frac{5}{2}=-\frac{5\sqrt{287}i}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{-5+5\sqrt{287}i}{2} x=\frac{-5\sqrt{287}i-5}{2}$$