Solve 2x-1/x+1+3x-1/x+2=x-7/x-1+4

$$\frac { 2 x - 1 } { x + 1 } + \frac { 3 x - 1 } { x + 2 } = \frac { x - 7 } { x - 1 } + 4$$

Solve for x

$$x = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

$$x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,-1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,x-1.

$$\left(x-1\right)\left(x+2\right)\left(2x-1\right)+\left(x^{2}-1\right)\left(3x-1\right)=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x-1 by x+2 and combine like terms.

$$\left(x^{2}+x-2\right)\left(2x-1\right)+\left(x^{2}-1\right)\left(3x-1\right)=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}+x-2 by 2x-1 and combine like terms.

$$2x^{3}+x^{2}-5x+2+\left(x^{2}-1\right)\left(3x-1\right)=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}-1 by 3x-1.

$$2x^{3}+x^{2}-5x+2+3x^{3}-x^{2}-3x+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

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

$$5x^{3}+x^{2}-5x+2-x^{2}-3x+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Combine x^{2} and -x^{2} to get 0.

$$5x^{3}-5x+2-3x+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Combine -5x and -3x to get -8x.

$$5x^{3}-8x+2+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Add 2 and 1 to get 3.

$$5x^{3}-8x+3=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x+1 by x+2 and combine like terms.

$$5x^{3}-8x+3=\left(x^{2}+3x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}+3x+2 by x-7 and combine like terms.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x-1 by x+1 and combine like terms.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+\left(x^{2}-1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}-1 by x+2.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+\left(x^{3}+2x^{2}-x-2\right)\times 4$$

Use the distributive property to multiply x^{3}+2x^{2}-x-2 by 4.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+4x^{3}+8x^{2}-4x-8$$

Combine x^{3} and 4x^{3} to get 5x^{3}.

$$5x^{3}-8x+3=5x^{3}-4x^{2}-19x-14+8x^{2}-4x-8$$

Combine -4x^{2} and 8x^{2} to get 4x^{2}.

$$5x^{3}-8x+3=5x^{3}+4x^{2}-19x-14-4x-8$$

Combine -19x and -4x to get -23x.

$$5x^{3}-8x+3=5x^{3}+4x^{2}-23x-14-8$$

Subtract 8 from -14 to get -22.

$$5x^{3}-8x+3=5x^{3}+4x^{2}-23x-22$$

Subtract 5x^{3} from both sides.

$$5x^{3}-8x+3-5x^{3}=4x^{2}-23x-22$$

Combine 5x^{3} and -5x^{3} to get 0.

$$-8x+3=4x^{2}-23x-22$$

Subtract 4x^{2} from both sides.

$$-8x+3-4x^{2}=-23x-22$$

Add 23x to both sides.

$$-8x+3-4x^{2}+23x=-22$$

Combine -8x and 23x to get 15x.

$$15x+3-4x^{2}=-22$$

Add 22 to both sides.

$$15x+3-4x^{2}+22=0$$

Add 3 and 22 to get 25.

$$15x+25-4x^{2}=0$$

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.

$$-4x^{2}+15x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 15 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-4\right)\times 25}}{2\left(-4\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-4\right)\times 25}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-15±\sqrt{225+16\times 25}}{2\left(-4\right)}$$

Multiply 16 times 25.

$$x=\frac{-15±\sqrt{225+400}}{2\left(-4\right)}$$

Add 225 to 400.

$$x=\frac{-15±\sqrt{625}}{2\left(-4\right)}$$

Take the square root of 625.

$$x=\frac{-15±25}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-15±25}{-8}$$

Now solve the equation x=\frac{-15±25}{-8} when ± is plus. Add -15 to 25.

$$x=\frac{10}{-8}$$

Reduce the fraction \frac{10}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{4}$$

Now solve the equation x=\frac{-15±25}{-8} when ± is minus. Subtract 25 from -15.

$$x=-\frac{40}{-8}$$

Divide -40 by -8.

$$x=5$$

The equation is now solved.

$$x=-\frac{5}{4} x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,-1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,x-1.

$$\left(x-1\right)\left(x+2\right)\left(2x-1\right)+\left(x^{2}-1\right)\left(3x-1\right)=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x-1 by x+2 and combine like terms.

$$\left(x^{2}+x-2\right)\left(2x-1\right)+\left(x^{2}-1\right)\left(3x-1\right)=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}+x-2 by 2x-1 and combine like terms.

$$2x^{3}+x^{2}-5x+2+\left(x^{2}-1\right)\left(3x-1\right)=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}-1 by 3x-1.

$$2x^{3}+x^{2}-5x+2+3x^{3}-x^{2}-3x+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

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

$$5x^{3}+x^{2}-5x+2-x^{2}-3x+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Combine x^{2} and -x^{2} to get 0.

$$5x^{3}-5x+2-3x+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Combine -5x and -3x to get -8x.

$$5x^{3}-8x+2+1=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Add 2 and 1 to get 3.

$$5x^{3}-8x+3=\left(x+1\right)\left(x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x+1 by x+2 and combine like terms.

$$5x^{3}-8x+3=\left(x^{2}+3x+2\right)\left(x-7\right)+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}+3x+2 by x-7 and combine like terms.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+\left(x-1\right)\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x-1 by x+1 and combine like terms.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+\left(x^{2}-1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x^{2}-1 by x+2.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+\left(x^{3}+2x^{2}-x-2\right)\times 4$$

Use the distributive property to multiply x^{3}+2x^{2}-x-2 by 4.

$$5x^{3}-8x+3=x^{3}-4x^{2}-19x-14+4x^{3}+8x^{2}-4x-8$$

Combine x^{3} and 4x^{3} to get 5x^{3}.

$$5x^{3}-8x+3=5x^{3}-4x^{2}-19x-14+8x^{2}-4x-8$$

Combine -4x^{2} and 8x^{2} to get 4x^{2}.

$$5x^{3}-8x+3=5x^{3}+4x^{2}-19x-14-4x-8$$

Combine -19x and -4x to get -23x.

$$5x^{3}-8x+3=5x^{3}+4x^{2}-23x-14-8$$

Subtract 8 from -14 to get -22.

$$5x^{3}-8x+3=5x^{3}+4x^{2}-23x-22$$

Subtract 5x^{3} from both sides.

$$5x^{3}-8x+3-5x^{3}=4x^{2}-23x-22$$

Combine 5x^{3} and -5x^{3} to get 0.

$$-8x+3=4x^{2}-23x-22$$

Subtract 4x^{2} from both sides.

$$-8x+3-4x^{2}=-23x-22$$

Add 23x to both sides.

$$-8x+3-4x^{2}+23x=-22$$

Combine -8x and 23x to get 15x.

$$15x+3-4x^{2}=-22$$

Subtract 3 from both sides.

$$15x-4x^{2}=-22-3$$

Subtract 3 from -22 to get -25.

$$15x-4x^{2}=-25$$

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.

$$-4x^{2}+15x=-25$$

Divide both sides by -4.

$$\frac{-4x^{2}+15x}{-4}=-\frac{25}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{15}{-4}x=-\frac{25}{-4}$$

Divide 15 by -4.

$$x^{2}-\frac{15}{4}x=-\frac{25}{-4}$$

Divide -25 by -4.

$$x^{2}-\frac{15}{4}x=\frac{25}{4}$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=\frac{25}{4}+\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{25}{4}+\frac{225}{64}$$

Add \frac{25}{4} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{625}{64}$$

Factor x^{2}-\frac{15}{4}x+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{625}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{625}{64}}$$

Simplify.

$$x-\frac{15}{8}=\frac{25}{8} x-\frac{15}{8}=-\frac{25}{8}$$

Add \frac{15}{8} to both sides of the equation.

$$x=5 x=-\frac{5}{4}$$