Solve 2/x+3+3/2=7/2x+1

$$\frac { 2 } { x + 3 } + \frac { 3 } { 2 } = \frac { 7 } { 2 x + 1 }$$

Solve for x

$$x=\frac{\sqrt{921}}{12}-\frac{5}{4}\approx 1.278998484$$

$$x=-\frac{\sqrt{921}}{12}-\frac{5}{4}\approx -3.778998484$$

Steps Using the Quadratic Formula

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

$$\left(4x+2\right)\times 2+2\left(x+3\right)\left(2x+1\right)\times \frac{3}{2}=\left(2x+6\right)\times 7$$

Use the distributive property to multiply 4x+2 by 2.

$$8x+4+2\left(x+3\right)\left(2x+1\right)\times \frac{3}{2}=\left(2x+6\right)\times 7$$

Multiply 2 and \frac{3}{2} to get 3.

$$8x+4+3\left(x+3\right)\left(2x+1\right)=\left(2x+6\right)\times 7$$

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

$$8x+4+\left(3x+9\right)\left(2x+1\right)=\left(2x+6\right)\times 7$$

Use the distributive property to multiply 3x+9 by 2x+1 and combine like terms.

$$8x+4+6x^{2}+21x+9=\left(2x+6\right)\times 7$$

Combine 8x and 21x to get 29x.

$$29x+4+6x^{2}+9=\left(2x+6\right)\times 7$$

Add 4 and 9 to get 13.

$$29x+13+6x^{2}=\left(2x+6\right)\times 7$$

Use the distributive property to multiply 2x+6 by 7.

$$29x+13+6x^{2}=14x+42$$

Subtract 14x from both sides.

$$29x+13+6x^{2}-14x=42$$

Combine 29x and -14x to get 15x.

$$15x+13+6x^{2}=42$$

Subtract 42 from both sides.

$$15x+13+6x^{2}-42=0$$

Subtract 42 from 13 to get -29.

$$15x-29+6x^{2}=0$$

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.

$$6x^{2}+15x-29=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 15 for b, and -29 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 6\left(-29\right)}}{2\times 6}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 6\left(-29\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-15±\sqrt{225-24\left(-29\right)}}{2\times 6}$$

Multiply -24 times -29.

$$x=\frac{-15±\sqrt{225+696}}{2\times 6}$$

Add 225 to 696.

$$x=\frac{-15±\sqrt{921}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-15±\sqrt{921}}{12}$$

Now solve the equation x=\frac{-15±\sqrt{921}}{12} when ± is plus. Add -15 to \sqrt{921}.

$$x=\frac{\sqrt{921}-15}{12}$$

Divide -15+\sqrt{921} by 12.

$$x=\frac{\sqrt{921}}{12}-\frac{5}{4}$$

Now solve the equation x=\frac{-15±\sqrt{921}}{12} when ± is minus. Subtract \sqrt{921} from -15.

$$x=\frac{-\sqrt{921}-15}{12}$$

Divide -15-\sqrt{921} by 12.

$$x=-\frac{\sqrt{921}}{12}-\frac{5}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{921}}{12}-\frac{5}{4} x=-\frac{\sqrt{921}}{12}-\frac{5}{4}$$

Steps for Completing the Square

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

$$\left(4x+2\right)\times 2+2\left(x+3\right)\left(2x+1\right)\times \frac{3}{2}=\left(2x+6\right)\times 7$$

Use the distributive property to multiply 4x+2 by 2.

$$8x+4+2\left(x+3\right)\left(2x+1\right)\times \frac{3}{2}=\left(2x+6\right)\times 7$$

Multiply 2 and \frac{3}{2} to get 3.

$$8x+4+3\left(x+3\right)\left(2x+1\right)=\left(2x+6\right)\times 7$$

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

$$8x+4+\left(3x+9\right)\left(2x+1\right)=\left(2x+6\right)\times 7$$

Use the distributive property to multiply 3x+9 by 2x+1 and combine like terms.

$$8x+4+6x^{2}+21x+9=\left(2x+6\right)\times 7$$

Combine 8x and 21x to get 29x.

$$29x+4+6x^{2}+9=\left(2x+6\right)\times 7$$

Add 4 and 9 to get 13.

$$29x+13+6x^{2}=\left(2x+6\right)\times 7$$

Use the distributive property to multiply 2x+6 by 7.

$$29x+13+6x^{2}=14x+42$$

Subtract 14x from both sides.

$$29x+13+6x^{2}-14x=42$$

Combine 29x and -14x to get 15x.

$$15x+13+6x^{2}=42$$

Subtract 13 from both sides.

$$15x+6x^{2}=42-13$$

Subtract 13 from 42 to get 29.

$$15x+6x^{2}=29$$

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.

$$6x^{2}+15x=29$$

Divide both sides by 6.

$$\frac{6x^{2}+15x}{6}=\frac{29}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{15}{6}x=\frac{29}{6}$$

Reduce the fraction \frac{15}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{5}{2}x=\frac{29}{6}$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\frac{29}{6}+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{29}{6}+\frac{25}{16}$$

Add \frac{29}{6} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{307}{48}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{307}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{307}{48}}$$

Simplify.

$$x+\frac{5}{4}=\frac{\sqrt{921}}{12} x+\frac{5}{4}=-\frac{\sqrt{921}}{12}$$

Subtract \frac{5}{4} from both sides of the equation.

$$x=\frac{\sqrt{921}}{12}-\frac{5}{4} x=-\frac{\sqrt{921}}{12}-\frac{5}{4}$$