Solve 5/3x+2-3/2x-3=4

$$\frac { 5 } { 3 x + 2 } - \frac { 3 } { 2 x - 3 } = 4$$

Solve for x

$$x=-\frac{1}{8}=-0.125$$

$$x=1$$

Steps Using the Quadratic Formula

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

$$\left(2x-3\right)\times 5-\left(3x+2\right)\times 3=4\left(2x-3\right)\left(3x+2\right)$$

Use the distributive property to multiply 2x-3 by 5.

$$10x-15-\left(3x+2\right)\times 3=4\left(2x-3\right)\left(3x+2\right)$$

Use the distributive property to multiply 3x+2 by 3.

$$10x-15-\left(9x+6\right)=4\left(2x-3\right)\left(3x+2\right)$$

To find the opposite of 9x+6, find the opposite of each term.

$$10x-15-9x-6=4\left(2x-3\right)\left(3x+2\right)$$

Combine 10x and -9x to get x.

$$x-15-6=4\left(2x-3\right)\left(3x+2\right)$$

Subtract 6 from -15 to get -21.

$$x-21=4\left(2x-3\right)\left(3x+2\right)$$

Use the distributive property to multiply 4 by 2x-3.

$$x-21=\left(8x-12\right)\left(3x+2\right)$$

Use the distributive property to multiply 8x-12 by 3x+2 and combine like terms.

$$x-21=24x^{2}-20x-24$$

Subtract 24x^{2} from both sides.

$$x-21-24x^{2}=-20x-24$$

Add 20x to both sides.

$$x-21-24x^{2}+20x=-24$$

Combine x and 20x to get 21x.

$$21x-21-24x^{2}=-24$$

Add 24 to both sides.

$$21x-21-24x^{2}+24=0$$

Add -21 and 24 to get 3.

$$21x+3-24x^{2}=0$$

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.

$$-24x^{2}+21x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, 21 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-24\right)\times 3}}{2\left(-24\right)}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\left(-24\right)\times 3}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-21±\sqrt{441+96\times 3}}{2\left(-24\right)}$$

Multiply 96 times 3.

$$x=\frac{-21±\sqrt{441+288}}{2\left(-24\right)}$$

Add 441 to 288.

$$x=\frac{-21±\sqrt{729}}{2\left(-24\right)}$$

Take the square root of 729.

$$x=\frac{-21±27}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-21±27}{-48}$$

Now solve the equation x=\frac{-21±27}{-48} when ± is plus. Add -21 to 27.

$$x=\frac{6}{-48}$$

Reduce the fraction \frac{6}{-48} to lowest terms by extracting and canceling out 6.

$$x=-\frac{1}{8}$$

Now solve the equation x=\frac{-21±27}{-48} when ± is minus. Subtract 27 from -21.

$$x=-\frac{48}{-48}$$

Divide -48 by -48.

$$x=1$$

The equation is now solved.

$$x=-\frac{1}{8} x=1$$

Steps for Completing the Square

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

$$\left(2x-3\right)\times 5-\left(3x+2\right)\times 3=4\left(2x-3\right)\left(3x+2\right)$$

Use the distributive property to multiply 2x-3 by 5.

$$10x-15-\left(3x+2\right)\times 3=4\left(2x-3\right)\left(3x+2\right)$$

Use the distributive property to multiply 3x+2 by 3.

$$10x-15-\left(9x+6\right)=4\left(2x-3\right)\left(3x+2\right)$$

To find the opposite of 9x+6, find the opposite of each term.

$$10x-15-9x-6=4\left(2x-3\right)\left(3x+2\right)$$

Combine 10x and -9x to get x.

$$x-15-6=4\left(2x-3\right)\left(3x+2\right)$$

Subtract 6 from -15 to get -21.

$$x-21=4\left(2x-3\right)\left(3x+2\right)$$

Use the distributive property to multiply 4 by 2x-3.

$$x-21=\left(8x-12\right)\left(3x+2\right)$$

Use the distributive property to multiply 8x-12 by 3x+2 and combine like terms.

$$x-21=24x^{2}-20x-24$$

Subtract 24x^{2} from both sides.

$$x-21-24x^{2}=-20x-24$$

Add 20x to both sides.

$$x-21-24x^{2}+20x=-24$$

Combine x and 20x to get 21x.

$$21x-21-24x^{2}=-24$$

Add 21 to both sides.

$$21x-24x^{2}=-24+21$$

Add -24 and 21 to get -3.

$$21x-24x^{2}=-3$$

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.

$$-24x^{2}+21x=-3$$

Divide both sides by -24.

$$\frac{-24x^{2}+21x}{-24}=-\frac{3}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$x^{2}+\frac{21}{-24}x=-\frac{3}{-24}$$

Reduce the fraction \frac{21}{-24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{7}{8}x=-\frac{3}{-24}$$

Reduce the fraction \frac{-3}{-24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{7}{8}x=\frac{1}{8}$$

Divide -\frac{7}{8}, the coefficient of the x term, by 2 to get -\frac{7}{16}. Then add the square of -\frac{7}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{8}x+\left(-\frac{7}{16}\right)^{2}=\frac{1}{8}+\left(-\frac{7}{16}\right)^{2}$$

Square -\frac{7}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=\frac{1}{8}+\frac{49}{256}$$

Add \frac{1}{8} to \frac{49}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=\frac{81}{256}$$

Factor x^{2}-\frac{7}{8}x+\frac{49}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{16}\right)^{2}=\frac{81}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{16}\right)^{2}}=\sqrt{\frac{81}{256}}$$

Simplify.

$$x-\frac{7}{16}=\frac{9}{16} x-\frac{7}{16}=-\frac{9}{16}$$

Add \frac{7}{16} to both sides of the equation.

$$x=1 x=-\frac{1}{8}$$