Solve 520div(x+10)+1=520+x

$$520 \div (x+10)+1=520+x$$

Solve for x

$$x=\frac{\sqrt{261161}-529}{2}\approx -8.980431278$$

$$x=\frac{-\sqrt{261161}-529}{2}\approx -520.019568722$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -10 since division by zero is not defined. Multiply both sides of the equation by x+10.

$$520+x+10=\left(x+10\right)\times 520+\left(x+10\right)x$$

Add 520 and 10 to get 530.

$$530+x=\left(x+10\right)\times 520+\left(x+10\right)x$$

Use the distributive property to multiply x+10 by 520.

$$530+x=520x+5200+\left(x+10\right)x$$

Use the distributive property to multiply x+10 by x.

$$530+x=520x+5200+x^{2}+10x$$

Combine 520x and 10x to get 530x.

$$530+x=530x+5200+x^{2}$$

Subtract 530x from both sides.

$$530+x-530x=5200+x^{2}$$

Combine x and -530x to get -529x.

$$530-529x=5200+x^{2}$$

Subtract 5200 from both sides.

$$530-529x-5200=x^{2}$$

Subtract 5200 from 530 to get -4670.

$$-4670-529x=x^{2}$$

Subtract x^{2} from both sides.

$$-4670-529x-x^{2}=0$$

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.

$$-x^{2}-529x-4670=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -529 for b, and -4670 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-529\right)±\sqrt{\left(-529\right)^{2}-4\left(-1\right)\left(-4670\right)}}{2\left(-1\right)}$$

Square -529.

$$x=\frac{-\left(-529\right)±\sqrt{279841-4\left(-1\right)\left(-4670\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-529\right)±\sqrt{279841+4\left(-4670\right)}}{2\left(-1\right)}$$

Multiply 4 times -4670.

$$x=\frac{-\left(-529\right)±\sqrt{279841-18680}}{2\left(-1\right)}$$

Add 279841 to -18680.

$$x=\frac{-\left(-529\right)±\sqrt{261161}}{2\left(-1\right)}$$

The opposite of -529 is 529.

$$x=\frac{529±\sqrt{261161}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{529±\sqrt{261161}}{-2}$$

Now solve the equation x=\frac{529±\sqrt{261161}}{-2} when ± is plus. Add 529 to \sqrt{261161}.

$$x=\frac{\sqrt{261161}+529}{-2}$$

Divide 529+\sqrt{261161} by -2.

$$x=\frac{-\sqrt{261161}-529}{2}$$

Now solve the equation x=\frac{529±\sqrt{261161}}{-2} when ± is minus. Subtract \sqrt{261161} from 529.

$$x=\frac{529-\sqrt{261161}}{-2}$$

Divide 529-\sqrt{261161} by -2.

$$x=\frac{\sqrt{261161}-529}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{261161}-529}{2} x=\frac{\sqrt{261161}-529}{2}$$

Steps for Completing the Square

Variable x cannot be equal to -10 since division by zero is not defined. Multiply both sides of the equation by x+10.

$$520+x+10=\left(x+10\right)\times 520+\left(x+10\right)x$$

Add 520 and 10 to get 530.

$$530+x=\left(x+10\right)\times 520+\left(x+10\right)x$$

Use the distributive property to multiply x+10 by 520.

$$530+x=520x+5200+\left(x+10\right)x$$

Use the distributive property to multiply x+10 by x.

$$530+x=520x+5200+x^{2}+10x$$

Combine 520x and 10x to get 530x.

$$530+x=530x+5200+x^{2}$$

Subtract 530x from both sides.

$$530+x-530x=5200+x^{2}$$

Combine x and -530x to get -529x.

$$530-529x=5200+x^{2}$$

Subtract x^{2} from both sides.

$$530-529x-x^{2}=5200$$

Subtract 530 from both sides.

$$-529x-x^{2}=5200-530$$

Subtract 530 from 5200 to get 4670.

$$-529x-x^{2}=4670$$

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.

$$-x^{2}-529x=4670$$

Divide both sides by -1.

$$\frac{-x^{2}-529x}{-1}=\frac{4670}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{529}{-1}\right)x=\frac{4670}{-1}$$

Divide -529 by -1.

$$x^{2}+529x=\frac{4670}{-1}$$

Divide 4670 by -1.

$$x^{2}+529x=-4670$$

Divide 529, the coefficient of the x term, by 2 to get \frac{529}{2}. Then add the square of \frac{529}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+529x+\left(\frac{529}{2}\right)^{2}=-4670+\left(\frac{529}{2}\right)^{2}$$

Square \frac{529}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+529x+\frac{279841}{4}=-4670+\frac{279841}{4}$$

Add -4670 to \frac{279841}{4}.

$$x^{2}+529x+\frac{279841}{4}=\frac{261161}{4}$$

Factor x^{2}+529x+\frac{279841}{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}.

$$\left(x+\frac{529}{2}\right)^{2}=\frac{261161}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{529}{2}\right)^{2}}=\sqrt{\frac{261161}{4}}$$

Simplify.

$$x+\frac{529}{2}=\frac{\sqrt{261161}}{2} x+\frac{529}{2}=-\frac{\sqrt{261161}}{2}$$

Subtract \frac{529}{2} from both sides of the equation.

$$x=\frac{\sqrt{261161}-529}{2} x=\frac{-\sqrt{261161}-529}{2}$$