Solve 160/x+5=160+x+60/x-10

$$\frac { 160 } { x } + 5 = \frac { 160 + x + 60 } { x - 10 }$$

Solve for x

$$x = \frac{5 \sqrt{377} + 55}{4} \approx 38.020609799$$

$$x=\frac{55-5\sqrt{377}}{4}\approx -10.520609799$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 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,x-10.

$$\left(x-10\right)\times 160+x\left(x-10\right)\times 5=x\left(160+x+60\right)$$

Use the distributive property to multiply x-10 by 160.

$$160x-1600+x\left(x-10\right)\times 5=x\left(160+x+60\right)$$

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

$$160x-1600+\left(x^{2}-10x\right)\times 5=x\left(160+x+60\right)$$

Use the distributive property to multiply x^{2}-10x by 5.

$$160x-1600+5x^{2}-50x=x\left(160+x+60\right)$$

Combine 160x and -50x to get 110x.

$$110x-1600+5x^{2}=x\left(160+x+60\right)$$

Add 160 and 60 to get 220.

$$110x-1600+5x^{2}=x\left(220+x\right)$$

Use the distributive property to multiply x by 220+x.

$$110x-1600+5x^{2}=220x+x^{2}$$

Subtract 220x from both sides.

$$110x-1600+5x^{2}-220x=x^{2}$$

Combine 110x and -220x to get -110x.

$$-110x-1600+5x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$-110x-1600+5x^{2}-x^{2}=0$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$-110x-1600+4x^{2}=0$$

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.

$$4x^{2}-110x-1600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -110 for b, and -1600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\times 4\left(-1600\right)}}{2\times 4}$$

Square -110.

$$x=\frac{-\left(-110\right)±\sqrt{12100-4\times 4\left(-1600\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-110\right)±\sqrt{12100-16\left(-1600\right)}}{2\times 4}$$

Multiply -16 times -1600.

$$x=\frac{-\left(-110\right)±\sqrt{12100+25600}}{2\times 4}$$

Add 12100 to 25600.

$$x=\frac{-\left(-110\right)±\sqrt{37700}}{2\times 4}$$

Take the square root of 37700.

$$x=\frac{-\left(-110\right)±10\sqrt{377}}{2\times 4}$$

The opposite of -110 is 110.

$$x=\frac{110±10\sqrt{377}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{110±10\sqrt{377}}{8}$$

Now solve the equation x=\frac{110±10\sqrt{377}}{8} when ± is plus. Add 110 to 10\sqrt{377}.

$$x=\frac{10\sqrt{377}+110}{8}$$

Divide 110+10\sqrt{377} by 8.

$$x=\frac{5\sqrt{377}+55}{4}$$

Now solve the equation x=\frac{110±10\sqrt{377}}{8} when ± is minus. Subtract 10\sqrt{377} from 110.

$$x=\frac{110-10\sqrt{377}}{8}$$

Divide 110-10\sqrt{377} by 8.

$$x=\frac{55-5\sqrt{377}}{4}$$

The equation is now solved.

$$x=\frac{5\sqrt{377}+55}{4} x=\frac{55-5\sqrt{377}}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 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,x-10.

$$\left(x-10\right)\times 160+x\left(x-10\right)\times 5=x\left(160+x+60\right)$$

Use the distributive property to multiply x-10 by 160.

$$160x-1600+x\left(x-10\right)\times 5=x\left(160+x+60\right)$$

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

$$160x-1600+\left(x^{2}-10x\right)\times 5=x\left(160+x+60\right)$$

Use the distributive property to multiply x^{2}-10x by 5.

$$160x-1600+5x^{2}-50x=x\left(160+x+60\right)$$

Combine 160x and -50x to get 110x.

$$110x-1600+5x^{2}=x\left(160+x+60\right)$$

Add 160 and 60 to get 220.

$$110x-1600+5x^{2}=x\left(220+x\right)$$

Use the distributive property to multiply x by 220+x.

$$110x-1600+5x^{2}=220x+x^{2}$$

Subtract 220x from both sides.

$$110x-1600+5x^{2}-220x=x^{2}$$

Combine 110x and -220x to get -110x.

$$-110x-1600+5x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$-110x-1600+5x^{2}-x^{2}=0$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$-110x-1600+4x^{2}=0$$

Add 1600 to both sides. Anything plus zero gives itself.

$$-110x+4x^{2}=1600$$

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.

$$4x^{2}-110x=1600$$

Divide both sides by 4.

$$\frac{4x^{2}-110x}{4}=\frac{1600}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{110}{4}\right)x=\frac{1600}{4}$$

Reduce the fraction \frac{-110}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{55}{2}x=\frac{1600}{4}$$

Divide 1600 by 4.

$$x^{2}-\frac{55}{2}x=400$$

Divide -\frac{55}{2}, the coefficient of the x term, by 2 to get -\frac{55}{4}. Then add the square of -\frac{55}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{55}{2}x+\left(-\frac{55}{4}\right)^{2}=400+\left(-\frac{55}{4}\right)^{2}$$

Square -\frac{55}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{55}{2}x+\frac{3025}{16}=400+\frac{3025}{16}$$

Add 400 to \frac{3025}{16}.

$$x^{2}-\frac{55}{2}x+\frac{3025}{16}=\frac{9425}{16}$$

Factor x^{2}-\frac{55}{2}x+\frac{3025}{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}.

$$\left(x-\frac{55}{4}\right)^{2}=\frac{9425}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{4}\right)^{2}}=\sqrt{\frac{9425}{16}}$$

Simplify.

$$x-\frac{55}{4}=\frac{5\sqrt{377}}{4} x-\frac{55}{4}=-\frac{5\sqrt{377}}{4}$$

Add \frac{55}{4} to both sides of the equation.

$$x=\frac{5\sqrt{377}+55}{4} x=\frac{55-5\sqrt{377}}{4}$$