Solve x+5/2x-3+x/2x-3=6x+1/3x-2=

$$\frac{ x+5 }{ 2x-3 } + \frac{ x }{ 2x-3 } = \frac{ 6x+1 }{ 3x-2 } =$$

Solve for x

$$x=\frac{\sqrt{561}}{12}+\frac{9}{4}\approx 4.223786547$$

$$x=-\frac{\sqrt{561}}{12}+\frac{9}{4}\approx 0.276213453$$

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.

$$\left(3x-2\right)\left(x+5\right)+\left(3x-2\right)x=\left(2x-3\right)\left(6x+1\right)$$

Use the distributive property to multiply 3x-2 by x+5 and combine like terms.

$$3x^{2}+13x-10+\left(3x-2\right)x=\left(2x-3\right)\left(6x+1\right)$$

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

$$3x^{2}+13x-10+3x^{2}-2x=\left(2x-3\right)\left(6x+1\right)$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}+13x-10-2x=\left(2x-3\right)\left(6x+1\right)$$

Combine 13x and -2x to get 11x.

$$6x^{2}+11x-10=\left(2x-3\right)\left(6x+1\right)$$

Use the distributive property to multiply 2x-3 by 6x+1 and combine like terms.

$$6x^{2}+11x-10=12x^{2}-16x-3$$

Subtract 12x^{2} from both sides.

$$6x^{2}+11x-10-12x^{2}=-16x-3$$

Combine 6x^{2} and -12x^{2} to get -6x^{2}.

$$-6x^{2}+11x-10=-16x-3$$

Add 16x to both sides.

$$-6x^{2}+11x-10+16x=-3$$

Combine 11x and 16x to get 27x.

$$-6x^{2}+27x-10=-3$$

Add 3 to both sides.

$$-6x^{2}+27x-10+3=0$$

Add -10 and 3 to get -7.

$$-6x^{2}+27x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 27 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-6\right)\left(-7\right)}}{2\left(-6\right)}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-6\right)\left(-7\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-27±\sqrt{729+24\left(-7\right)}}{2\left(-6\right)}$$

Multiply 24 times -7.

$$x=\frac{-27±\sqrt{729-168}}{2\left(-6\right)}$$

Add 729 to -168.

$$x=\frac{-27±\sqrt{561}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-27±\sqrt{561}}{-12}$$

Now solve the equation x=\frac{-27±\sqrt{561}}{-12} when ± is plus. Add -27 to \sqrt{561}.

$$x=\frac{\sqrt{561}-27}{-12}$$

Divide -27+\sqrt{561} by -12.

$$x=-\frac{\sqrt{561}}{12}+\frac{9}{4}$$

Now solve the equation x=\frac{-27±\sqrt{561}}{-12} when ± is minus. Subtract \sqrt{561} from -27.

$$x=\frac{-\sqrt{561}-27}{-12}$$

Divide -27-\sqrt{561} by -12.

$$x=\frac{\sqrt{561}}{12}+\frac{9}{4}$$

The equation is now solved.

$$x=-\frac{\sqrt{561}}{12}+\frac{9}{4} x=\frac{\sqrt{561}}{12}+\frac{9}{4}$$

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.

$$\left(3x-2\right)\left(x+5\right)+\left(3x-2\right)x=\left(2x-3\right)\left(6x+1\right)$$

Use the distributive property to multiply 3x-2 by x+5 and combine like terms.

$$3x^{2}+13x-10+\left(3x-2\right)x=\left(2x-3\right)\left(6x+1\right)$$

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

$$3x^{2}+13x-10+3x^{2}-2x=\left(2x-3\right)\left(6x+1\right)$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}+13x-10-2x=\left(2x-3\right)\left(6x+1\right)$$

Combine 13x and -2x to get 11x.

$$6x^{2}+11x-10=\left(2x-3\right)\left(6x+1\right)$$

Use the distributive property to multiply 2x-3 by 6x+1 and combine like terms.

$$6x^{2}+11x-10=12x^{2}-16x-3$$

Subtract 12x^{2} from both sides.

$$6x^{2}+11x-10-12x^{2}=-16x-3$$

Combine 6x^{2} and -12x^{2} to get -6x^{2}.

$$-6x^{2}+11x-10=-16x-3$$

Add 16x to both sides.

$$-6x^{2}+11x-10+16x=-3$$

Combine 11x and 16x to get 27x.

$$-6x^{2}+27x-10=-3$$

Add 10 to both sides.

$$-6x^{2}+27x=-3+10$$

Add -3 and 10 to get 7.

$$-6x^{2}+27x=7$$

Divide both sides by -6.

$$\frac{-6x^{2}+27x}{-6}=\frac{7}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{27}{-6}x=\frac{7}{-6}$$

Reduce the fraction \frac{27}{-6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{9}{2}x=\frac{7}{-6}$$

Divide 7 by -6.

$$x^{2}-\frac{9}{2}x=-\frac{7}{6}$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-\frac{7}{6}+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{7}{6}+\frac{81}{16}$$

Add -\frac{7}{6} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{187}{48}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{187}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{187}{48}}$$

Simplify.

$$x-\frac{9}{4}=\frac{\sqrt{561}}{12} x-\frac{9}{4}=-\frac{\sqrt{561}}{12}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{\sqrt{561}}{12}+\frac{9}{4} x=-\frac{\sqrt{561}}{12}+\frac{9}{4}$$