Solve 560div(x+10)+1=560divx

$$560 \div (x+10)+1=560 \div x$$

Solve for x

$$x=-80$$

$$x=70$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x+10,x.

$$x\times 560+x\left(x+10\right)=\left(x+10\right)\times 560$$

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

$$x\times 560+x^{2}+10x=\left(x+10\right)\times 560$$

Combine x\times 560 and 10x to get 570x.

$$570x+x^{2}=\left(x+10\right)\times 560$$

Use the distributive property to multiply x+10 by 560.

$$570x+x^{2}=560x+5600$$

Subtract 560x from both sides.

$$570x+x^{2}-560x=5600$$

Combine 570x and -560x to get 10x.

$$10x+x^{2}=5600$$

Subtract 5600 from both sides.

$$10x+x^{2}-5600=0$$

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.

$$x^{2}+10x-5600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -5600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-5600\right)}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-5600\right)}}{2}$$

Multiply -4 times -5600.

$$x=\frac{-10±\sqrt{100+22400}}{2}$$

Add 100 to 22400.

$$x=\frac{-10±\sqrt{22500}}{2}$$

Take the square root of 22500.

$$x=\frac{-10±150}{2}$$

Now solve the equation x=\frac{-10±150}{2} when ± is plus. Add -10 to 150.

$$x=\frac{140}{2}$$

Divide 140 by 2.

$$x=70$$

Now solve the equation x=\frac{-10±150}{2} when ± is minus. Subtract 150 from -10.

$$x=-\frac{160}{2}$$

Divide -160 by 2.

$$x=-80$$

The equation is now solved.

$$x=70 x=-80$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x+10,x.

$$x\times 560+x\left(x+10\right)=\left(x+10\right)\times 560$$

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

$$x\times 560+x^{2}+10x=\left(x+10\right)\times 560$$

Combine x\times 560 and 10x to get 570x.

$$570x+x^{2}=\left(x+10\right)\times 560$$

Use the distributive property to multiply x+10 by 560.

$$570x+x^{2}=560x+5600$$

Subtract 560x from both sides.

$$570x+x^{2}-560x=5600$$

Combine 570x and -560x to get 10x.

$$10x+x^{2}=5600$$

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.

$$x^{2}+10x=5600$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=5600+5^{2}$$

Square 5.

$$x^{2}+10x+25=5600+25$$

Add 5600 to 25.

$$x^{2}+10x+25=5625$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{5625}$$

Simplify.

$$x+5=75 x+5=-75$$

Subtract 5 from both sides of the equation.

$$x=70 x=-80$$