Solve 24x-18/3x-2-8x-10/2x-3=2

$$\frac{ 24x-18 }{ 3x-2 } - \frac{ 8x-10 }{ 2x-3 } =2$$

Solve for x

$$x=\frac{\sqrt{15}}{6}+\frac{3}{2}\approx 2.145497224$$

$$x=-\frac{\sqrt{15}}{6}+\frac{3}{2}\approx 0.854502776$$

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.

$$\left(2x-3\right)\left(24x-18\right)-\left(3x-2\right)\left(8x-10\right)=2\left(2x-3\right)\left(3x-2\right)$$

Use the distributive property to multiply 2x-3 by 24x-18 and combine like terms.

$$48x^{2}-108x+54-\left(3x-2\right)\left(8x-10\right)=2\left(2x-3\right)\left(3x-2\right)$$

Use the distributive property to multiply 3x-2 by 8x-10 and combine like terms.

$$48x^{2}-108x+54-\left(24x^{2}-46x+20\right)=2\left(2x-3\right)\left(3x-2\right)$$

To find the opposite of 24x^{2}-46x+20, find the opposite of each term.

$$48x^{2}-108x+54-24x^{2}+46x-20=2\left(2x-3\right)\left(3x-2\right)$$

Combine 48x^{2} and -24x^{2} to get 24x^{2}.

$$24x^{2}-108x+54+46x-20=2\left(2x-3\right)\left(3x-2\right)$$

Combine -108x and 46x to get -62x.

$$24x^{2}-62x+54-20=2\left(2x-3\right)\left(3x-2\right)$$

Subtract 20 from 54 to get 34.

$$24x^{2}-62x+34=2\left(2x-3\right)\left(3x-2\right)$$

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

$$24x^{2}-62x+34=\left(4x-6\right)\left(3x-2\right)$$

Use the distributive property to multiply 4x-6 by 3x-2 and combine like terms.

$$24x^{2}-62x+34=12x^{2}-26x+12$$

Subtract 12x^{2} from both sides.

$$24x^{2}-62x+34-12x^{2}=-26x+12$$

Combine 24x^{2} and -12x^{2} to get 12x^{2}.

$$12x^{2}-62x+34=-26x+12$$

Add 26x to both sides.

$$12x^{2}-62x+34+26x=12$$

Combine -62x and 26x to get -36x.

$$12x^{2}-36x+34=12$$

Subtract 12 from both sides.

$$12x^{2}-36x+34-12=0$$

Subtract 12 from 34 to get 22.

$$12x^{2}-36x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -36 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 12\times 22}}{2\times 12}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 12\times 22}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-36\right)±\sqrt{1296-48\times 22}}{2\times 12}$$

Multiply -48 times 22.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1056}}{2\times 12}$$

Add 1296 to -1056.

$$x=\frac{-\left(-36\right)±\sqrt{240}}{2\times 12}$$

Take the square root of 240.

$$x=\frac{-\left(-36\right)±4\sqrt{15}}{2\times 12}$$

The opposite of -36 is 36.

$$x=\frac{36±4\sqrt{15}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{36±4\sqrt{15}}{24}$$

Now solve the equation x=\frac{36±4\sqrt{15}}{24} when ± is plus. Add 36 to 4\sqrt{15}.

$$x=\frac{4\sqrt{15}+36}{24}$$

Divide 36+4\sqrt{15} by 24.

$$x=\frac{\sqrt{15}}{6}+\frac{3}{2}$$

Now solve the equation x=\frac{36±4\sqrt{15}}{24} when ± is minus. Subtract 4\sqrt{15} from 36.

$$x=\frac{36-4\sqrt{15}}{24}$$

Divide 36-4\sqrt{15} by 24.

$$x=-\frac{\sqrt{15}}{6}+\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{15}}{6}+\frac{3}{2} x=-\frac{\sqrt{15}}{6}+\frac{3}{2}$$

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.

$$\left(2x-3\right)\left(24x-18\right)-\left(3x-2\right)\left(8x-10\right)=2\left(2x-3\right)\left(3x-2\right)$$

Use the distributive property to multiply 2x-3 by 24x-18 and combine like terms.

$$48x^{2}-108x+54-\left(3x-2\right)\left(8x-10\right)=2\left(2x-3\right)\left(3x-2\right)$$

Use the distributive property to multiply 3x-2 by 8x-10 and combine like terms.

$$48x^{2}-108x+54-\left(24x^{2}-46x+20\right)=2\left(2x-3\right)\left(3x-2\right)$$

To find the opposite of 24x^{2}-46x+20, find the opposite of each term.

$$48x^{2}-108x+54-24x^{2}+46x-20=2\left(2x-3\right)\left(3x-2\right)$$

Combine 48x^{2} and -24x^{2} to get 24x^{2}.

$$24x^{2}-108x+54+46x-20=2\left(2x-3\right)\left(3x-2\right)$$

Combine -108x and 46x to get -62x.

$$24x^{2}-62x+54-20=2\left(2x-3\right)\left(3x-2\right)$$

Subtract 20 from 54 to get 34.

$$24x^{2}-62x+34=2\left(2x-3\right)\left(3x-2\right)$$

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

$$24x^{2}-62x+34=\left(4x-6\right)\left(3x-2\right)$$

Use the distributive property to multiply 4x-6 by 3x-2 and combine like terms.

$$24x^{2}-62x+34=12x^{2}-26x+12$$

Subtract 12x^{2} from both sides.

$$24x^{2}-62x+34-12x^{2}=-26x+12$$

Combine 24x^{2} and -12x^{2} to get 12x^{2}.

$$12x^{2}-62x+34=-26x+12$$

Add 26x to both sides.

$$12x^{2}-62x+34+26x=12$$

Combine -62x and 26x to get -36x.

$$12x^{2}-36x+34=12$$

Subtract 34 from both sides.

$$12x^{2}-36x=12-34$$

Subtract 34 from 12 to get -22.

$$12x^{2}-36x=-22$$

Divide both sides by 12.

$$\frac{12x^{2}-36x}{12}=-\frac{22}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{36}{12}\right)x=-\frac{22}{12}$$

Divide -36 by 12.

$$x^{2}-3x=-\frac{22}{12}$$

Reduce the fraction \frac{-22}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-3x=-\frac{11}{6}$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-\frac{11}{6}+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-\frac{11}{6}+\frac{9}{4}$$

Add -\frac{11}{6} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=\frac{5}{12}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{5}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{5}{12}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{15}}{6} x-\frac{3}{2}=-\frac{\sqrt{15}}{6}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{15}}{6}+\frac{3}{2} x=-\frac{\sqrt{15}}{6}+\frac{3}{2}$$