Solve x^2/x+20=10

$$\frac { x ^ { 2 } } { x + 20 } = 10$$

Solve for x

$$x=-10$$

$$x=20$$

Steps Using Factoring

Variable x cannot be equal to -20 since division by zero is not defined. Multiply both sides of the equation by x+20.

$$x^{2}=10\left(x+20\right)$$

Use the distributive property to multiply 10 by x+20.

$$x^{2}=10x+200$$

Subtract 10x from both sides.

$$x^{2}-10x=200$$

Subtract 200 from both sides.

$$x^{2}-10x-200=0$$

To solve the equation, factor x^{2}-10x-200 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.

$$a+b=-10 ab=-200$$

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 -200.

$$1,-200 2,-100 4,-50 5,-40 8,-25 10,-20$$

Calculate the sum for each pair.

$$1-200=-199 2-100=-98 4-50=-46 5-40=-35 8-25=-17 10-20=-10$$

The solution is the pair that gives sum -10.

$$a=-20 b=10$$

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

$$\left(x-20\right)\left(x+10\right)$$

To find equation solutions, solve x-20=0 and x+10=0.

$$x=20 x=-10$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -20 since division by zero is not defined. Multiply both sides of the equation by x+20.

$$x^{2}=10\left(x+20\right)$$

Use the distributive property to multiply 10 by x+20.

$$x^{2}=10x+200$$

Subtract 10x from both sides.

$$x^{2}-10x=200$$

Subtract 200 from both sides.

$$x^{2}-10x-200=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-200. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\left(-200\right)=-200$$

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 -200.

$$1,-200 2,-100 4,-50 5,-40 8,-25 10,-20$$

Calculate the sum for each pair.

$$1-200=-199 2-100=-98 4-50=-46 5-40=-35 8-25=-17 10-20=-10$$

The solution is the pair that gives sum -10.

$$a=-20 b=10$$

Rewrite x^{2}-10x-200 as \left(x^{2}-20x\right)+\left(10x-200\right).

$$\left(x^{2}-20x\right)+\left(10x-200\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(x-20\right)+10\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(x+10\right)$$

To find equation solutions, solve x-20=0 and x+10=0.

$$x=20 x=-10$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -20 since division by zero is not defined. Multiply both sides of the equation by x+20.

$$x^{2}=10\left(x+20\right)$$

Use the distributive property to multiply 10 by x+20.

$$x^{2}=10x+200$$

Subtract 10x from both sides.

$$x^{2}-10x=200$$

Subtract 200 from both sides.

$$x^{2}-10x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-200\right)}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-200\right)}}{2}$$

Multiply -4 times -200.

$$x=\frac{-\left(-10\right)±\sqrt{100+800}}{2}$$

Add 100 to 800.

$$x=\frac{-\left(-10\right)±\sqrt{900}}{2}$$

Take the square root of 900.

$$x=\frac{-\left(-10\right)±30}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±30}{2}$$

Now solve the equation x=\frac{10±30}{2} when ± is plus. Add 10 to 30.

$$x=\frac{40}{2}$$

Divide 40 by 2.

$$x=20$$

Now solve the equation x=\frac{10±30}{2} when ± is minus. Subtract 30 from 10.

$$x=-\frac{20}{2}$$

Divide -20 by 2.

$$x=-10$$

The equation is now solved.

$$x=20 x=-10$$

Steps for Completing the Square

Variable x cannot be equal to -20 since division by zero is not defined. Multiply both sides of the equation by x+20.

$$x^{2}=10\left(x+20\right)$$

Use the distributive property to multiply 10 by x+20.

$$x^{2}=10x+200$$

Subtract 10x from both sides.

$$x^{2}-10x=200$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=200+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=200+25$$

Add 200 to 25.

$$x^{2}-10x+25=225$$

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}.

$$\left(x-5\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-5=15 x-5=-15$$

Add 5 to both sides of the equation.

$$x=20 x=-10$$