Solve 2x+1/x-2+x+7/x=5

$$\frac { 2 x + 1 } { x - 2 } + \frac { x + 7 } { x } = 5$$

Solve for x

$$x=1$$

$$x=7$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\left(2x+1\right)+\left(x-2\right)\left(x+7\right)=5x\left(x-2\right)$$

Use the distributive property to multiply x by 2x+1.

$$2x^{2}+x+\left(x-2\right)\left(x+7\right)=5x\left(x-2\right)$$

Use the distributive property to multiply x-2 by x+7 and combine like terms.

$$2x^{2}+x+x^{2}+5x-14=5x\left(x-2\right)$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}+x+5x-14=5x\left(x-2\right)$$

Combine x and 5x to get 6x.

$$3x^{2}+6x-14=5x\left(x-2\right)$$

Use the distributive property to multiply 5x by x-2.

$$3x^{2}+6x-14=5x^{2}-10x$$

Subtract 5x^{2} from both sides.

$$3x^{2}+6x-14-5x^{2}=-10x$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$-2x^{2}+6x-14=-10x$$

Add 10x to both sides.

$$-2x^{2}+6x-14+10x=0$$

Combine 6x and 10x to get 16x.

$$-2x^{2}+16x-14=0$$

Divide both sides by 2.

$$-x^{2}+8x-7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-7\right)=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=7 b=1$$

Rewrite -x^{2}+8x-7 as \left(-x^{2}+7x\right)+\left(x-7\right).

$$\left(-x^{2}+7x\right)+\left(x-7\right)$$

Factor out -x in -x^{2}+7x.

$$-x\left(x-7\right)+x-7$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x+1\right)$$

To find equation solutions, solve x-7=0 and -x+1=0.

$$x=7 x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\left(2x+1\right)+\left(x-2\right)\left(x+7\right)=5x\left(x-2\right)$$

Use the distributive property to multiply x by 2x+1.

$$2x^{2}+x+\left(x-2\right)\left(x+7\right)=5x\left(x-2\right)$$

Use the distributive property to multiply x-2 by x+7 and combine like terms.

$$2x^{2}+x+x^{2}+5x-14=5x\left(x-2\right)$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}+x+5x-14=5x\left(x-2\right)$$

Combine x and 5x to get 6x.

$$3x^{2}+6x-14=5x\left(x-2\right)$$

Use the distributive property to multiply 5x by x-2.

$$3x^{2}+6x-14=5x^{2}-10x$$

Subtract 5x^{2} from both sides.

$$3x^{2}+6x-14-5x^{2}=-10x$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$-2x^{2}+6x-14=-10x$$

Add 10x to both sides.

$$-2x^{2}+6x-14+10x=0$$

Combine 6x and 10x to get 16x.

$$-2x^{2}+16x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 16 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-2\right)\left(-14\right)}}{2\left(-2\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-2\right)\left(-14\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-16±\sqrt{256+8\left(-14\right)}}{2\left(-2\right)}$$

Multiply 8 times -14.

$$x=\frac{-16±\sqrt{256-112}}{2\left(-2\right)}$$

Add 256 to -112.

$$x=\frac{-16±\sqrt{144}}{2\left(-2\right)}$$

Take the square root of 144.

$$x=\frac{-16±12}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-16±12}{-4}$$

Now solve the equation x=\frac{-16±12}{-4} when ± is plus. Add -16 to 12.

$$x=-\frac{4}{-4}$$

Divide -4 by -4.

$$x=1$$

Now solve the equation x=\frac{-16±12}{-4} when ± is minus. Subtract 12 from -16.

$$x=-\frac{28}{-4}$$

Divide -28 by -4.

$$x=7$$

The equation is now solved.

$$x=1 x=7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\left(2x+1\right)+\left(x-2\right)\left(x+7\right)=5x\left(x-2\right)$$

Use the distributive property to multiply x by 2x+1.

$$2x^{2}+x+\left(x-2\right)\left(x+7\right)=5x\left(x-2\right)$$

Use the distributive property to multiply x-2 by x+7 and combine like terms.

$$2x^{2}+x+x^{2}+5x-14=5x\left(x-2\right)$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}+x+5x-14=5x\left(x-2\right)$$

Combine x and 5x to get 6x.

$$3x^{2}+6x-14=5x\left(x-2\right)$$

Use the distributive property to multiply 5x by x-2.

$$3x^{2}+6x-14=5x^{2}-10x$$

Subtract 5x^{2} from both sides.

$$3x^{2}+6x-14-5x^{2}=-10x$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$-2x^{2}+6x-14=-10x$$

Add 10x to both sides.

$$-2x^{2}+6x-14+10x=0$$

Combine 6x and 10x to get 16x.

$$-2x^{2}+16x-14=0$$

Add 14 to both sides. Anything plus zero gives itself.

$$-2x^{2}+16x=14$$

Divide both sides by -2.

$$\frac{-2x^{2}+16x}{-2}=\frac{14}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{16}{-2}x=\frac{14}{-2}$$

Divide 16 by -2.

$$x^{2}-8x=\frac{14}{-2}$$

Divide 14 by -2.

$$x^{2}-8x=-7$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-7+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-7+16$$

Add -7 to 16.

$$x^{2}-8x+16=9$$

Factor x^{2}-8x+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}.

$$\left(x-4\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-4=3 x-4=-3$$

Add 4 to both sides of the equation.

$$x=7 x=1$$