Solve 2(3x+2)/x-1=3x+4/x+2

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

Solve for x

$$x=-4$$

$$x=-1$$

Steps Using the Quadratic Formula

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

$$\left(x+2\right)\times 2\left(3x+2\right)=\left(x-1\right)\left(3x+4\right)$$

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

$$\left(2x+4\right)\left(3x+2\right)=\left(x-1\right)\left(3x+4\right)$$

Use the distributive property to multiply 2x+4 by 3x+2 and combine like terms.

$$6x^{2}+16x+8=\left(x-1\right)\left(3x+4\right)$$

Use the distributive property to multiply x-1 by 3x+4 and combine like terms.

$$6x^{2}+16x+8=3x^{2}+x-4$$

Subtract 3x^{2} from both sides.

$$6x^{2}+16x+8-3x^{2}=x-4$$

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

$$3x^{2}+16x+8=x-4$$

Subtract x from both sides.

$$3x^{2}+16x+8-x=-4$$

Combine 16x and -x to get 15x.

$$3x^{2}+15x+8=-4$$

Add 4 to both sides.

$$3x^{2}+15x+8+4=0$$

Add 8 and 4 to get 12.

$$3x^{2}+15x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 15 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 3\times 12}}{2\times 3}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-15±\sqrt{225-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$x=\frac{-15±\sqrt{225-144}}{2\times 3}$$

Add 225 to -144.

$$x=\frac{-15±\sqrt{81}}{2\times 3}$$

Take the square root of 81.

$$x=\frac{-15±9}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-15±9}{6}$$

Now solve the equation x=\frac{-15±9}{6} when ± is plus. Add -15 to 9.

$$x=-\frac{6}{6}$$

Divide -6 by 6.

$$x=-1$$

Now solve the equation x=\frac{-15±9}{6} when ± is minus. Subtract 9 from -15.

$$x=-\frac{24}{6}$$

Divide -24 by 6.

$$x=-4$$

The equation is now solved.

$$x=-1 x=-4$$

Steps for Completing the Square

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

$$\left(x+2\right)\times 2\left(3x+2\right)=\left(x-1\right)\left(3x+4\right)$$

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

$$\left(2x+4\right)\left(3x+2\right)=\left(x-1\right)\left(3x+4\right)$$

Use the distributive property to multiply 2x+4 by 3x+2 and combine like terms.

$$6x^{2}+16x+8=\left(x-1\right)\left(3x+4\right)$$

Use the distributive property to multiply x-1 by 3x+4 and combine like terms.

$$6x^{2}+16x+8=3x^{2}+x-4$$

Subtract 3x^{2} from both sides.

$$6x^{2}+16x+8-3x^{2}=x-4$$

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

$$3x^{2}+16x+8=x-4$$

Subtract x from both sides.

$$3x^{2}+16x+8-x=-4$$

Combine 16x and -x to get 15x.

$$3x^{2}+15x+8=-4$$

Subtract 8 from both sides.

$$3x^{2}+15x=-4-8$$

Subtract 8 from -4 to get -12.

$$3x^{2}+15x=-12$$

Divide both sides by 3.

$$\frac{3x^{2}+15x}{3}=-\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{15}{3}x=-\frac{12}{3}$$

Divide 15 by 3.

$$x^{2}+5x=-\frac{12}{3}$$

Divide -12 by 3.

$$x^{2}+5x=-4$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-4+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=-4+\frac{25}{4}$$

Add -4 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{9}{4}$$

Factor x^{2}+5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{3}{2} x+\frac{5}{2}=-\frac{3}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=-1 x=-4$$