Solve (x+3)/(x-2)-8/3=(x+2)/(x-1)

$$\frac { ( x + 3 ) } { ( x - 2 ) } - \frac { 8 } { 3 } = \frac { ( x + 2 ) } { ( x - 1 ) }$$

Solve for x

$$x = \frac{13}{4} = 3\frac{1}{4} = 3.25$$

$$x=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-2\right)\left(x-1\right), the least common multiple of x-2,3,x-1.

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

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

$$3x^{2}+6x-9+3\left(x-2\right)\left(x-1\right)\left(-\frac{8}{3}\right)=\left(3x-6\right)\left(x+2\right)$$

Multiply 3 and -\frac{8}{3} to get -8.

$$3x^{2}+6x-9-8\left(x-2\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply -8 by x-2.

$$3x^{2}+6x-9+\left(-8x+16\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply -8x+16 by x-1 and combine like terms.

$$3x^{2}+6x-9-8x^{2}+24x-16=\left(3x-6\right)\left(x+2\right)$$

Combine 3x^{2} and -8x^{2} to get -5x^{2}.

$$-5x^{2}+6x-9+24x-16=\left(3x-6\right)\left(x+2\right)$$

Combine 6x and 24x to get 30x.

$$-5x^{2}+30x-9-16=\left(3x-6\right)\left(x+2\right)$$

Subtract 16 from -9 to get -25.

$$-5x^{2}+30x-25=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$-5x^{2}+30x-25=3x^{2}-12$$

Subtract 3x^{2} from both sides.

$$-5x^{2}+30x-25-3x^{2}=-12$$

Combine -5x^{2} and -3x^{2} to get -8x^{2}.

$$-8x^{2}+30x-25=-12$$

Add 12 to both sides.

$$-8x^{2}+30x-25+12=0$$

Add -25 and 12 to get -13.

$$-8x^{2}+30x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 30 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-8\right)\left(-13\right)}}{2\left(-8\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-8\right)\left(-13\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-30±\sqrt{900+32\left(-13\right)}}{2\left(-8\right)}$$

Multiply 32 times -13.

$$x=\frac{-30±\sqrt{900-416}}{2\left(-8\right)}$$

Add 900 to -416.

$$x=\frac{-30±\sqrt{484}}{2\left(-8\right)}$$

Take the square root of 484.

$$x=\frac{-30±22}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-30±22}{-16}$$

Now solve the equation x=\frac{-30±22}{-16} when ± is plus. Add -30 to 22.

$$x=-\frac{8}{-16}$$

Reduce the fraction \frac{-8}{-16} to lowest terms by extracting and canceling out 8.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{-30±22}{-16} when ± is minus. Subtract 22 from -30.

$$x=-\frac{52}{-16}$$

Reduce the fraction \frac{-52}{-16} to lowest terms by extracting and canceling out 4.

$$x=\frac{13}{4}$$

The equation is now solved.

$$x=\frac{1}{2} x=\frac{13}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-2\right)\left(x-1\right), the least common multiple of x-2,3,x-1.

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

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

$$3x^{2}+6x-9+3\left(x-2\right)\left(x-1\right)\left(-\frac{8}{3}\right)=\left(3x-6\right)\left(x+2\right)$$

Multiply 3 and -\frac{8}{3} to get -8.

$$3x^{2}+6x-9-8\left(x-2\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply -8 by x-2.

$$3x^{2}+6x-9+\left(-8x+16\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply -8x+16 by x-1 and combine like terms.

$$3x^{2}+6x-9-8x^{2}+24x-16=\left(3x-6\right)\left(x+2\right)$$

Combine 3x^{2} and -8x^{2} to get -5x^{2}.

$$-5x^{2}+6x-9+24x-16=\left(3x-6\right)\left(x+2\right)$$

Combine 6x and 24x to get 30x.

$$-5x^{2}+30x-9-16=\left(3x-6\right)\left(x+2\right)$$

Subtract 16 from -9 to get -25.

$$-5x^{2}+30x-25=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$-5x^{2}+30x-25=3x^{2}-12$$

Subtract 3x^{2} from both sides.

$$-5x^{2}+30x-25-3x^{2}=-12$$

Combine -5x^{2} and -3x^{2} to get -8x^{2}.

$$-8x^{2}+30x-25=-12$$

Add 25 to both sides.

$$-8x^{2}+30x=-12+25$$

Add -12 and 25 to get 13.

$$-8x^{2}+30x=13$$

Divide both sides by -8.

$$\frac{-8x^{2}+30x}{-8}=\frac{13}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{30}{-8}x=\frac{13}{-8}$$

Reduce the fraction \frac{30}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{4}x=\frac{13}{-8}$$

Divide 13 by -8.

$$x^{2}-\frac{15}{4}x=-\frac{13}{8}$$

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.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=-\frac{13}{8}+\left(-\frac{15}{8}\right)^{2}$$

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

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=-\frac{13}{8}+\frac{225}{64}$$

Add -\frac{13}{8} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{121}{64}$$

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}.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$x-\frac{15}{8}=\frac{11}{8} x-\frac{15}{8}=-\frac{11}{8}$$

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

$$x=\frac{13}{4} x=\frac{1}{2}$$