Solve 2/3+1/4x-1/12(x+1)=0

$$\frac { 2 } { 3 } + \frac { 1 } { 4 x } - \frac { 1 } { 12 ( x + 1 ) } = 0$$

Solve for x

$$x=-\frac{3}{4}=-0.75$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+1\right), the least common multiple of 3,4x,12\left(x+1\right).

$$12x\left(x+1\right)\times \frac{2}{3}+3x+3-x=0$$

Use the distributive property to multiply 12x by x+1.

$$\left(12x^{2}+12x\right)\times \frac{2}{3}+3x+3-x=0$$

Use the distributive property to multiply 12x^{2}+12x by \frac{2}{3}.

$$8x^{2}+8x+3x+3-x=0$$

Combine 8x and 3x to get 11x.

$$8x^{2}+11x+3-x=0$$

Combine 11x and -x to get 10x.

$$8x^{2}+10x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=10 ab=8\times 3=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 10.

$$a=4 b=6$$

Rewrite 8x^{2}+10x+3 as \left(8x^{2}+4x\right)+\left(6x+3\right).

$$\left(8x^{2}+4x\right)+\left(6x+3\right)$$

Factor out 4x in the first and 3 in the second group.

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

Factor out common term 2x+1 by using distributive property.

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

To find equation solutions, solve 2x+1=0 and 4x+3=0.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+1\right), the least common multiple of 3,4x,12\left(x+1\right).

$$12x\left(x+1\right)\times \frac{2}{3}+3x+3-x=0$$

Use the distributive property to multiply 12x by x+1.

$$\left(12x^{2}+12x\right)\times \frac{2}{3}+3x+3-x=0$$

Use the distributive property to multiply 12x^{2}+12x by \frac{2}{3}.

$$8x^{2}+8x+3x+3-x=0$$

Combine 8x and 3x to get 11x.

$$8x^{2}+11x+3-x=0$$

Combine 11x and -x to get 10x.

$$8x^{2}+10x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 10 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 8\times 3}}{2\times 8}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 8\times 3}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-10±\sqrt{100-32\times 3}}{2\times 8}$$

Multiply -32 times 3.

$$x=\frac{-10±\sqrt{100-96}}{2\times 8}$$

Add 100 to -96.

$$x=\frac{-10±\sqrt{4}}{2\times 8}$$

Take the square root of 4.

$$x=\frac{-10±2}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-10±2}{16}$$

Now solve the equation x=\frac{-10±2}{16} when ± is plus. Add -10 to 2.

$$x=-\frac{8}{16}$$

Reduce the fraction \frac{-8}{16} to lowest terms by extracting and canceling out 8.

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

Now solve the equation x=\frac{-10±2}{16} when ± is minus. Subtract 2 from -10.

$$x=-\frac{12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{4}$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+1\right), the least common multiple of 3,4x,12\left(x+1\right).

$$12x\left(x+1\right)\times \frac{2}{3}+3x+3-x=0$$

Use the distributive property to multiply 12x by x+1.

$$\left(12x^{2}+12x\right)\times \frac{2}{3}+3x+3-x=0$$

Use the distributive property to multiply 12x^{2}+12x by \frac{2}{3}.

$$8x^{2}+8x+3x+3-x=0$$

Combine 8x and 3x to get 11x.

$$8x^{2}+11x+3-x=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$8x^{2}+11x-x=-3$$

Combine 11x and -x to get 10x.

$$8x^{2}+10x=-3$$

Divide both sides by 8.

$$\frac{8x^{2}+10x}{8}=-\frac{3}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{10}{8}x=-\frac{3}{8}$$

Reduce the fraction \frac{10}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{4}x=-\frac{3}{8}$$

Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-\frac{3}{8}+\left(\frac{5}{8}\right)^{2}$$

Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{3}{8}+\frac{25}{64}$$

Add -\frac{3}{8} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=\frac{1}{64}$$

Factor x^{2}+\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$x+\frac{5}{8}=\frac{1}{8} x+\frac{5}{8}=-\frac{1}{8}$$

Subtract \frac{5}{8} from both sides of the equation.

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