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

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

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x-1\right), the least common multiple of x-1,4x.

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

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

$$8x^{2}+12x=\left(x-1\right)\left(2x+3\right)$$

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

$$8x^{2}+12x=2x^{2}+x-3$$

Subtract 2x^{2} from both sides.

$$8x^{2}+12x-2x^{2}=x-3$$

Combine 8x^{2} and -2x^{2} to get 6x^{2}.

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

Subtract x from both sides.

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

Combine 12x and -x to get 11x.

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

Add 3 to both sides.

$$6x^{2}+11x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 11 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 6\times 3}}{2\times 6}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 6\times 3}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-11±\sqrt{121-24\times 3}}{2\times 6}$$

Multiply -24 times 3.

$$x=\frac{-11±\sqrt{121-72}}{2\times 6}$$

Add 121 to -72.

$$x=\frac{-11±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$x=\frac{-11±7}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-11±7}{12}$$

Now solve the equation x=\frac{-11±7}{12} when ± is plus. Add -11 to 7.

$$x=-\frac{4}{12}$$

Reduce the fraction \frac{-4}{12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{3}$$

Now solve the equation x=\frac{-11±7}{12} when ± is minus. Subtract 7 from -11.

$$x=-\frac{18}{12}$$

Reduce the fraction \frac{-18}{12} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x-1\right), the least common multiple of x-1,4x.

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

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

$$8x^{2}+12x=\left(x-1\right)\left(2x+3\right)$$

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

$$8x^{2}+12x=2x^{2}+x-3$$

Subtract 2x^{2} from both sides.

$$8x^{2}+12x-2x^{2}=x-3$$

Combine 8x^{2} and -2x^{2} to get 6x^{2}.

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

Subtract x from both sides.

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

Combine 12x and -x to get 11x.

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

Divide both sides by 6.

$$\frac{6x^{2}+11x}{6}=-\frac{3}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{11}{6}x=-\frac{3}{6}$$

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

$$x^{2}+\frac{11}{6}x=-\frac{1}{2}$$

Divide \frac{11}{6}, the coefficient of the x term, by 2 to get \frac{11}{12}. Then add the square of \frac{11}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{6}x+\left(\frac{11}{12}\right)^{2}=-\frac{1}{2}+\left(\frac{11}{12}\right)^{2}$$

Square \frac{11}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=-\frac{1}{2}+\frac{121}{144}$$

Add -\frac{1}{2} to \frac{121}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{49}{144}$$

Factor x^{2}+\frac{11}{6}x+\frac{121}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x+\frac{11}{12}=\frac{7}{12} x+\frac{11}{12}=-\frac{7}{12}$$

Subtract \frac{11}{12} from both sides of the equation.

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