Solve 3/2x-3+3/x+1=4

$$\frac { 3 } { 2 x - 3 } + \frac { 3 } { x + 1 } = 4$$

Solve for x

$$x=-\frac{3}{8}=-0.375$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$\left(x+1\right)\times 3+\left(2x-3\right)\times 3=4\left(2x-3\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 3.

$$3x+3+\left(2x-3\right)\times 3=4\left(2x-3\right)\left(x+1\right)$$

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

$$3x+3+6x-9=4\left(2x-3\right)\left(x+1\right)$$

Combine 3x and 6x to get 9x.

$$9x+3-9=4\left(2x-3\right)\left(x+1\right)$$

Subtract 9 from 3 to get -6.

$$9x-6=4\left(2x-3\right)\left(x+1\right)$$

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

$$9x-6=\left(8x-12\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-12 by x+1 and combine like terms.

$$9x-6=8x^{2}-4x-12$$

Subtract 8x^{2} from both sides.

$$9x-6-8x^{2}=-4x-12$$

Add 4x to both sides.

$$9x-6-8x^{2}+4x=-12$$

Combine 9x and 4x to get 13x.

$$13x-6-8x^{2}=-12$$

Add 12 to both sides.

$$13x-6-8x^{2}+12=0$$

Add -6 and 12 to get 6.

$$13x+6-8x^{2}=0$$

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.

$$-8x^{2}+13x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 13 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-8\right)\times 6}}{2\left(-8\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-8\right)\times 6}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-13±\sqrt{169+32\times 6}}{2\left(-8\right)}$$

Multiply 32 times 6.

$$x=\frac{-13±\sqrt{169+192}}{2\left(-8\right)}$$

Add 169 to 192.

$$x=\frac{-13±\sqrt{361}}{2\left(-8\right)}$$

Take the square root of 361.

$$x=\frac{-13±19}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-13±19}{-16}$$

Now solve the equation x=\frac{-13±19}{-16} when ± is plus. Add -13 to 19.

$$x=\frac{6}{-16}$$

Reduce the fraction \frac{6}{-16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{8}$$

Now solve the equation x=\frac{-13±19}{-16} when ± is minus. Subtract 19 from -13.

$$x=-\frac{32}{-16}$$

Divide -32 by -16.

$$x=2$$

The equation is now solved.

$$x=-\frac{3}{8} x=2$$

Steps for Completing the Square

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

$$\left(x+1\right)\times 3+\left(2x-3\right)\times 3=4\left(2x-3\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 3.

$$3x+3+\left(2x-3\right)\times 3=4\left(2x-3\right)\left(x+1\right)$$

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

$$3x+3+6x-9=4\left(2x-3\right)\left(x+1\right)$$

Combine 3x and 6x to get 9x.

$$9x+3-9=4\left(2x-3\right)\left(x+1\right)$$

Subtract 9 from 3 to get -6.

$$9x-6=4\left(2x-3\right)\left(x+1\right)$$

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

$$9x-6=\left(8x-12\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-12 by x+1 and combine like terms.

$$9x-6=8x^{2}-4x-12$$

Subtract 8x^{2} from both sides.

$$9x-6-8x^{2}=-4x-12$$

Add 4x to both sides.

$$9x-6-8x^{2}+4x=-12$$

Combine 9x and 4x to get 13x.

$$13x-6-8x^{2}=-12$$

Add 6 to both sides.

$$13x-8x^{2}=-12+6$$

Add -12 and 6 to get -6.

$$13x-8x^{2}=-6$$

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.

$$-8x^{2}+13x=-6$$

Divide both sides by -8.

$$\frac{-8x^{2}+13x}{-8}=-\frac{6}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{13}{-8}x=-\frac{6}{-8}$$

Divide 13 by -8.

$$x^{2}-\frac{13}{8}x=-\frac{6}{-8}$$

Reduce the fraction \frac{-6}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{13}{8}x=\frac{3}{4}$$

Divide -\frac{13}{8}, the coefficient of the x term, by 2 to get -\frac{13}{16}. Then add the square of -\frac{13}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{8}x+\left(-\frac{13}{16}\right)^{2}=\frac{3}{4}+\left(-\frac{13}{16}\right)^{2}$$

Square -\frac{13}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{8}x+\frac{169}{256}=\frac{3}{4}+\frac{169}{256}$$

Add \frac{3}{4} to \frac{169}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{8}x+\frac{169}{256}=\frac{361}{256}$$

Factor x^{2}-\frac{13}{8}x+\frac{169}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{16}\right)^{2}=\frac{361}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{16}\right)^{2}}=\sqrt{\frac{361}{256}}$$

Simplify.

$$x-\frac{13}{16}=\frac{19}{16} x-\frac{13}{16}=-\frac{19}{16}$$

Add \frac{13}{16} to both sides of the equation.

$$x=2 x=-\frac{3}{8}$$