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

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

Solve for x

$$x=1$$

$$x = \frac{13}{7} = 1\frac{6}{7} \approx 1.857142857$$

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 2x-3,3x-2.

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

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

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

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

$$3x^{2}-2x+10x-15=4\left(2x-3\right)\left(3x-2\right)$$

Combine -2x and 10x to get 8x.

$$3x^{2}+8x-15=4\left(2x-3\right)\left(3x-2\right)$$

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

$$3x^{2}+8x-15=\left(8x-12\right)\left(3x-2\right)$$

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

$$3x^{2}+8x-15=24x^{2}-52x+24$$

Subtract 24x^{2} from both sides.

$$3x^{2}+8x-15-24x^{2}=-52x+24$$

Combine 3x^{2} and -24x^{2} to get -21x^{2}.

$$-21x^{2}+8x-15=-52x+24$$

Add 52x to both sides.

$$-21x^{2}+8x-15+52x=24$$

Combine 8x and 52x to get 60x.

$$-21x^{2}+60x-15=24$$

Subtract 24 from both sides.

$$-21x^{2}+60x-15-24=0$$

Subtract 24 from -15 to get -39.

$$-21x^{2}+60x-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -21 for a, 60 for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-21\right)\left(-39\right)}}{2\left(-21\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-21\right)\left(-39\right)}}{2\left(-21\right)}$$

Multiply -4 times -21.

$$x=\frac{-60±\sqrt{3600+84\left(-39\right)}}{2\left(-21\right)}$$

Multiply 84 times -39.

$$x=\frac{-60±\sqrt{3600-3276}}{2\left(-21\right)}$$

Add 3600 to -3276.

$$x=\frac{-60±\sqrt{324}}{2\left(-21\right)}$$

Take the square root of 324.

$$x=\frac{-60±18}{2\left(-21\right)}$$

Multiply 2 times -21.

$$x=\frac{-60±18}{-42}$$

Now solve the equation x=\frac{-60±18}{-42} when ± is plus. Add -60 to 18.

$$x=-\frac{42}{-42}$$

Divide -42 by -42.

$$x=1$$

Now solve the equation x=\frac{-60±18}{-42} when ± is minus. Subtract 18 from -60.

$$x=-\frac{78}{-42}$$

Reduce the fraction \frac{-78}{-42} to lowest terms by extracting and canceling out 6.

$$x=\frac{13}{7}$$

The equation is now solved.

$$x=1 x=\frac{13}{7}$$

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 2x-3,3x-2.

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

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

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

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

$$3x^{2}-2x+10x-15=4\left(2x-3\right)\left(3x-2\right)$$

Combine -2x and 10x to get 8x.

$$3x^{2}+8x-15=4\left(2x-3\right)\left(3x-2\right)$$

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

$$3x^{2}+8x-15=\left(8x-12\right)\left(3x-2\right)$$

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

$$3x^{2}+8x-15=24x^{2}-52x+24$$

Subtract 24x^{2} from both sides.

$$3x^{2}+8x-15-24x^{2}=-52x+24$$

Combine 3x^{2} and -24x^{2} to get -21x^{2}.

$$-21x^{2}+8x-15=-52x+24$$

Add 52x to both sides.

$$-21x^{2}+8x-15+52x=24$$

Combine 8x and 52x to get 60x.

$$-21x^{2}+60x-15=24$$

Add 15 to both sides.

$$-21x^{2}+60x=24+15$$

Add 24 and 15 to get 39.

$$-21x^{2}+60x=39$$

Divide both sides by -21.

$$\frac{-21x^{2}+60x}{-21}=\frac{39}{-21}$$

Dividing by -21 undoes the multiplication by -21.

$$x^{2}+\frac{60}{-21}x=\frac{39}{-21}$$

Reduce the fraction \frac{60}{-21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{20}{7}x=\frac{39}{-21}$$

Reduce the fraction \frac{39}{-21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{20}{7}x=-\frac{13}{7}$$

Divide -\frac{20}{7}, the coefficient of the x term, by 2 to get -\frac{10}{7}. Then add the square of -\frac{10}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{20}{7}x+\left(-\frac{10}{7}\right)^{2}=-\frac{13}{7}+\left(-\frac{10}{7}\right)^{2}$$

Square -\frac{10}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{20}{7}x+\frac{100}{49}=-\frac{13}{7}+\frac{100}{49}$$

Add -\frac{13}{7} to \frac{100}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{20}{7}x+\frac{100}{49}=\frac{9}{49}$$

Factor x^{2}-\frac{20}{7}x+\frac{100}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{10}{7}\right)^{2}=\frac{9}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{10}{7}\right)^{2}}=\sqrt{\frac{9}{49}}$$

Simplify.

$$x-\frac{10}{7}=\frac{3}{7} x-\frac{10}{7}=-\frac{3}{7}$$

Add \frac{10}{7} to both sides of the equation.

$$x=\frac{13}{7} x=1$$