Solve x^2/x+100=50

$$\frac { x ^ { 2 } } { x + 100 } = 50$$

Solve for x

$$x=-50$$

$$x=100$$

Steps Using Factoring

Variable x cannot be equal to -100 since division by zero is not defined. Multiply both sides of the equation by x+100.

$$x^{2}=50\left(x+100\right)$$

Use the distributive property to multiply 50 by x+100.

$$x^{2}=50x+5000$$

Subtract 50x from both sides.

$$x^{2}-50x=5000$$

Subtract 5000 from both sides.

$$x^{2}-50x-5000=0$$

To solve the equation, factor x^{2}-50x-5000 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-50 ab=-5000$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -5000.

$$1,-5000 2,-2500 4,-1250 5,-1000 8,-625 10,-500 20,-250 25,-200 40,-125 50,-100$$

Calculate the sum for each pair.

$$1-5000=-4999 2-2500=-2498 4-1250=-1246 5-1000=-995 8-625=-617 10-500=-490 20-250=-230 25-200=-175 40-125=-85 50-100=-50$$

The solution is the pair that gives sum -50.

$$a=-100 b=50$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-100\right)\left(x+50\right)$$

To find equation solutions, solve x-100=0 and x+50=0.

$$x=100 x=-50$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -100 since division by zero is not defined. Multiply both sides of the equation by x+100.

$$x^{2}=50\left(x+100\right)$$

Use the distributive property to multiply 50 by x+100.

$$x^{2}=50x+5000$$

Subtract 50x from both sides.

$$x^{2}-50x=5000$$

Subtract 5000 from both sides.

$$x^{2}-50x-5000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-5000. To find a and b, set up a system to be solved.

$$a+b=-50 ab=1\left(-5000\right)=-5000$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -5000.

$$1,-5000 2,-2500 4,-1250 5,-1000 8,-625 10,-500 20,-250 25,-200 40,-125 50,-100$$

Calculate the sum for each pair.

$$1-5000=-4999 2-2500=-2498 4-1250=-1246 5-1000=-995 8-625=-617 10-500=-490 20-250=-230 25-200=-175 40-125=-85 50-100=-50$$

The solution is the pair that gives sum -50.

$$a=-100 b=50$$

Rewrite x^{2}-50x-5000 as \left(x^{2}-100x\right)+\left(50x-5000\right).

$$\left(x^{2}-100x\right)+\left(50x-5000\right)$$

Factor out x in the first and 50 in the second group.

$$x\left(x-100\right)+50\left(x-100\right)$$

Factor out common term x-100 by using distributive property.

$$\left(x-100\right)\left(x+50\right)$$

To find equation solutions, solve x-100=0 and x+50=0.

$$x=100 x=-50$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -100 since division by zero is not defined. Multiply both sides of the equation by x+100.

$$x^{2}=50\left(x+100\right)$$

Use the distributive property to multiply 50 by x+100.

$$x^{2}=50x+5000$$

Subtract 50x from both sides.

$$x^{2}-50x=5000$$

Subtract 5000 from both sides.

$$x^{2}-50x-5000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -50 for b, and -5000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-5000\right)}}{2}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-5000\right)}}{2}$$

Multiply -4 times -5000.

$$x=\frac{-\left(-50\right)±\sqrt{2500+20000}}{2}$$

Add 2500 to 20000.

$$x=\frac{-\left(-50\right)±\sqrt{22500}}{2}$$

Take the square root of 22500.

$$x=\frac{-\left(-50\right)±150}{2}$$

The opposite of -50 is 50.

$$x=\frac{50±150}{2}$$

Now solve the equation x=\frac{50±150}{2} when ± is plus. Add 50 to 150.

$$x=\frac{200}{2}$$

Divide 200 by 2.

$$x=100$$

Now solve the equation x=\frac{50±150}{2} when ± is minus. Subtract 150 from 50.

$$x=-\frac{100}{2}$$

Divide -100 by 2.

$$x=-50$$

The equation is now solved.

$$x=100 x=-50$$

Steps for Completing the Square

Variable x cannot be equal to -100 since division by zero is not defined. Multiply both sides of the equation by x+100.

$$x^{2}=50\left(x+100\right)$$

Use the distributive property to multiply 50 by x+100.

$$x^{2}=50x+5000$$

Subtract 50x from both sides.

$$x^{2}-50x=5000$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=5000+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=5000+625$$

Add 5000 to 625.

$$x^{2}-50x+625=5625$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{5625}$$

Simplify.

$$x-25=75 x-25=-75$$

Add 25 to both sides of the equation.

$$x=100 x=-50$$