Solve 3/2x+1-1/3x+2=2,g

$$\frac { 3 } { 2 x + 1 } - \frac { 1 } { 3 x + 2 } = 2 , g$$

Solve for x

$$x=\frac{\sqrt{97}-7}{24}\approx 0.118702408$$

$$x=\frac{-\sqrt{97}-7}{24}\approx -0.702035742$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{3},-\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x+1\right)\left(3x+2\right), the least common multiple of 2x+1,3x+2.

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

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

$$9x+6-\left(2x+1\right)=2\left(2x+1\right)\left(3x+2\right)$$

To find the opposite of 2x+1, find the opposite of each term.

$$9x+6-2x-1=2\left(2x+1\right)\left(3x+2\right)$$

Combine 9x and -2x to get 7x.

$$7x+6-1=2\left(2x+1\right)\left(3x+2\right)$$

Subtract 1 from 6 to get 5.

$$7x+5=2\left(2x+1\right)\left(3x+2\right)$$

Use the distributive property to multiply 2 by 2x+1.

$$7x+5=\left(4x+2\right)\left(3x+2\right)$$

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

$$7x+5=12x^{2}+14x+4$$

Subtract 12x^{2} from both sides.

$$7x+5-12x^{2}=14x+4$$

Subtract 14x from both sides.

$$7x+5-12x^{2}-14x=4$$

Combine 7x and -14x to get -7x.

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

Subtract 4 from both sides.

$$-7x+5-12x^{2}-4=0$$

Subtract 4 from 5 to get 1.

$$-7x+1-12x^{2}=0$$

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.

$$-12x^{2}-7x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -7 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-12\right)}}{2\left(-12\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-12\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-\left(-7\right)±\sqrt{49+48}}{2\left(-12\right)}$$

Add 49 to 48.

$$x=\frac{-\left(-7\right)±\sqrt{97}}{2\left(-12\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{97}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{7±\sqrt{97}}{-24}$$

Now solve the equation x=\frac{7±\sqrt{97}}{-24} when ± is plus. Add 7 to \sqrt{97}.

$$x=\frac{\sqrt{97}+7}{-24}$$

Divide 7+\sqrt{97} by -24.

$$x=\frac{-\sqrt{97}-7}{24}$$

Now solve the equation x=\frac{7±\sqrt{97}}{-24} when ± is minus. Subtract \sqrt{97} from 7.

$$x=\frac{7-\sqrt{97}}{-24}$$

Divide 7-\sqrt{97} by -24.

$$x=\frac{\sqrt{97}-7}{24}$$

The equation is now solved.

$$x=\frac{-\sqrt{97}-7}{24} x=\frac{\sqrt{97}-7}{24}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{3},-\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x+1\right)\left(3x+2\right), the least common multiple of 2x+1,3x+2.

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

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

$$9x+6-\left(2x+1\right)=2\left(2x+1\right)\left(3x+2\right)$$

To find the opposite of 2x+1, find the opposite of each term.

$$9x+6-2x-1=2\left(2x+1\right)\left(3x+2\right)$$

Combine 9x and -2x to get 7x.

$$7x+6-1=2\left(2x+1\right)\left(3x+2\right)$$

Subtract 1 from 6 to get 5.

$$7x+5=2\left(2x+1\right)\left(3x+2\right)$$

Use the distributive property to multiply 2 by 2x+1.

$$7x+5=\left(4x+2\right)\left(3x+2\right)$$

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

$$7x+5=12x^{2}+14x+4$$

Subtract 12x^{2} from both sides.

$$7x+5-12x^{2}=14x+4$$

Subtract 14x from both sides.

$$7x+5-12x^{2}-14x=4$$

Combine 7x and -14x to get -7x.

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

Subtract 5 from both sides.

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

Subtract 5 from 4 to get -1.

$$-7x-12x^{2}=-1$$

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.

$$-12x^{2}-7x=-1$$

Divide both sides by -12.

$$\frac{-12x^{2}-7x}{-12}=-\frac{1}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$x^{2}+\left(-\frac{7}{-12}\right)x=-\frac{1}{-12}$$

Divide -7 by -12.

$$x^{2}+\frac{7}{12}x=-\frac{1}{-12}$$

Divide -1 by -12.

$$x^{2}+\frac{7}{12}x=\frac{1}{12}$$

Divide \frac{7}{12}, the coefficient of the x term, by 2 to get \frac{7}{24}. Then add the square of \frac{7}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{12}x+\left(\frac{7}{24}\right)^{2}=\frac{1}{12}+\left(\frac{7}{24}\right)^{2}$$

Square \frac{7}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{12}x+\frac{49}{576}=\frac{1}{12}+\frac{49}{576}$$

Add \frac{1}{12} to \frac{49}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{12}x+\frac{49}{576}=\frac{97}{576}$$

Factor x^{2}+\frac{7}{12}x+\frac{49}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{24}\right)^{2}=\frac{97}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{24}\right)^{2}}=\sqrt{\frac{97}{576}}$$

Simplify.

$$x+\frac{7}{24}=\frac{\sqrt{97}}{24} x+\frac{7}{24}=-\frac{\sqrt{97}}{24}$$

Subtract \frac{7}{24} from both sides of the equation.

$$x=\frac{\sqrt{97}-7}{24} x=\frac{-\sqrt{97}-7}{24}$$