Solve 3x-8/x-2=5x-2/x+5

$$\frac { 3 x - 8 } { x - 2 } = \frac { 5 x - 2 } { x + 5 }$$

Solve for x

$$x=4$$

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5.

$$\left(x+5\right)\left(3x-8\right)=\left(x-2\right)\left(5x-2\right)$$

Use the distributive property to multiply x+5 by 3x-8 and combine like terms.

$$3x^{2}+7x-40=\left(x-2\right)\left(5x-2\right)$$

Use the distributive property to multiply x-2 by 5x-2 and combine like terms.

$$3x^{2}+7x-40=5x^{2}-12x+4$$

Subtract 5x^{2} from both sides.

$$3x^{2}+7x-40-5x^{2}=-12x+4$$

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

$$-2x^{2}+7x-40=-12x+4$$

Add 12x to both sides.

$$-2x^{2}+7x-40+12x=4$$

Combine 7x and 12x to get 19x.

$$-2x^{2}+19x-40=4$$

Subtract 4 from both sides.

$$-2x^{2}+19x-40-4=0$$

Subtract 4 from -40 to get -44.

$$-2x^{2}+19x-44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 19 for b, and -44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-19±\sqrt{361+8\left(-44\right)}}{2\left(-2\right)}$$

Multiply 8 times -44.

$$x=\frac{-19±\sqrt{361-352}}{2\left(-2\right)}$$

Add 361 to -352.

$$x=\frac{-19±\sqrt{9}}{2\left(-2\right)}$$

Take the square root of 9.

$$x=\frac{-19±3}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-19±3}{-4}$$

Now solve the equation x=\frac{-19±3}{-4} when ± is plus. Add -19 to 3.

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

Now solve the equation x=\frac{-19±3}{-4} when ± is minus. Subtract 3 from -19.

$$x=-\frac{22}{-4}$$

Reduce the fraction \frac{-22}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{11}{2}$$

The equation is now solved.

$$x=4 x=\frac{11}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5.

$$\left(x+5\right)\left(3x-8\right)=\left(x-2\right)\left(5x-2\right)$$

Use the distributive property to multiply x+5 by 3x-8 and combine like terms.

$$3x^{2}+7x-40=\left(x-2\right)\left(5x-2\right)$$

Use the distributive property to multiply x-2 by 5x-2 and combine like terms.

$$3x^{2}+7x-40=5x^{2}-12x+4$$

Subtract 5x^{2} from both sides.

$$3x^{2}+7x-40-5x^{2}=-12x+4$$

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

$$-2x^{2}+7x-40=-12x+4$$

Add 12x to both sides.

$$-2x^{2}+7x-40+12x=4$$

Combine 7x and 12x to get 19x.

$$-2x^{2}+19x-40=4$$

Add 40 to both sides.

$$-2x^{2}+19x=4+40$$

Add 4 and 40 to get 44.

$$-2x^{2}+19x=44$$

Divide both sides by -2.

$$\frac{-2x^{2}+19x}{-2}=\frac{44}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{19}{-2}x=\frac{44}{-2}$$

Divide 19 by -2.

$$x^{2}-\frac{19}{2}x=\frac{44}{-2}$$

Divide 44 by -2.

$$x^{2}-\frac{19}{2}x=-22$$

Divide -\frac{19}{2}, the coefficient of the x term, by 2 to get -\frac{19}{4}. Then add the square of -\frac{19}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{2}x+\left(-\frac{19}{4}\right)^{2}=-22+\left(-\frac{19}{4}\right)^{2}$$

Square -\frac{19}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=-22+\frac{361}{16}$$

Add -22 to \frac{361}{16}.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=\frac{9}{16}$$

Factor x^{2}-\frac{19}{2}x+\frac{361}{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}.

$$\left(x-\frac{19}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$x-\frac{19}{4}=\frac{3}{4} x-\frac{19}{4}=-\frac{3}{4}$$

Add \frac{19}{4} to both sides of the equation.

$$x=\frac{11}{2} x=4$$