Solve x-1/x+2=10/3x-2

$$\frac { x - 1 } { x + 2 } = \frac { 10 } { 3 x - 2 }$$

Solve for x

$$x=-1$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-2\right)\left(x+2\right), the least common multiple of x+2,3x-2.

$$\left(3x-2\right)\left(x-1\right)=\left(x+2\right)\times 10$$

Use the distributive property to multiply 3x-2 by x-1 and combine like terms.

$$3x^{2}-5x+2=\left(x+2\right)\times 10$$

Use the distributive property to multiply x+2 by 10.

$$3x^{2}-5x+2=10x+20$$

Subtract 10x from both sides.

$$3x^{2}-5x+2-10x=20$$

Combine -5x and -10x to get -15x.

$$3x^{2}-15x+2=20$$

Subtract 20 from both sides.

$$3x^{2}-15x+2-20=0$$

Subtract 20 from 2 to get -18.

$$3x^{2}-15x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -15 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\left(-18\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\left(-18\right)}}{2\times 3}$$

Multiply -12 times -18.

$$x=\frac{-\left(-15\right)±\sqrt{225+216}}{2\times 3}$$

Add 225 to 216.

$$x=\frac{-\left(-15\right)±\sqrt{441}}{2\times 3}$$

Take the square root of 441.

$$x=\frac{-\left(-15\right)±21}{2\times 3}$$

The opposite of -15 is 15.

$$x=\frac{15±21}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{15±21}{6}$$

Now solve the equation x=\frac{15±21}{6} when ± is plus. Add 15 to 21.

$$x=\frac{36}{6}$$

Divide 36 by 6.

$$x=6$$

Now solve the equation x=\frac{15±21}{6} when ± is minus. Subtract 21 from 15.

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

Divide -6 by 6.

$$x=-1$$

The equation is now solved.

$$x=6 x=-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-2\right)\left(x+2\right), the least common multiple of x+2,3x-2.

$$\left(3x-2\right)\left(x-1\right)=\left(x+2\right)\times 10$$

Use the distributive property to multiply 3x-2 by x-1 and combine like terms.

$$3x^{2}-5x+2=\left(x+2\right)\times 10$$

Use the distributive property to multiply x+2 by 10.

$$3x^{2}-5x+2=10x+20$$

Subtract 10x from both sides.

$$3x^{2}-5x+2-10x=20$$

Combine -5x and -10x to get -15x.

$$3x^{2}-15x+2=20$$

Subtract 2 from both sides.

$$3x^{2}-15x=20-2$$

Subtract 2 from 20 to get 18.

$$3x^{2}-15x=18$$

Divide both sides by 3.

$$\frac{3x^{2}-15x}{3}=\frac{18}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{15}{3}\right)x=\frac{18}{3}$$

Divide -15 by 3.

$$x^{2}-5x=\frac{18}{3}$$

Divide 18 by 3.

$$x^{2}-5x=6$$

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.

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

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

$$x^{2}-5x+\frac{25}{4}=6+\frac{25}{4}$$

Add 6 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{49}{4}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

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

Add \frac{5}{2} to both sides of the equation.

$$x=6 x=-1$$