Solve 1/2x(x/3-1/2)+1/3x(frac{1}{4}-frac{1}{3})=frac{2x+1}{6}

$$\frac { 1 } { 2 } x ( \frac { x } { 3 } - \frac { 1 } { 2 } ) + \frac { 1 } { 3 } x ( \frac { 1 } { 4 } - \frac { 1 } { 3 } ) = \frac { 2 x + 1 } { 6 }$$

Solve for x

$$x = \frac{\sqrt{157} + 11}{6} \approx 3.921660681$$

$$x=\frac{11-\sqrt{157}}{6}\approx -0.254994014$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 2,3,4,6.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x}{3} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{3}{3}.

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

Since \frac{2x}{6} and \frac{3}{6} have the same denominator, subtract them by subtracting their numerators.

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

Express 6\times \frac{2x-3}{6} as a single fraction.

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

Cancel out 6 and 6.

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

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

$$2x^{2}-3x+4x\left(\frac{1}{4}-\frac{1}{3}\right)=2\left(2x+1\right)$$

Least common multiple of 4 and 3 is 12. Convert \frac{1}{4} and \frac{1}{3} to fractions with denominator 12.

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

Since \frac{3}{12} and \frac{4}{12} have the same denominator, subtract them by subtracting their numerators.

$$2x^{2}-3x+4x\times \frac{3-4}{12}=2\left(2x+1\right)$$

Subtract 4 from 3 to get -1.

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

Express 4\left(-\frac{1}{12}\right) as a single fraction.

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

Multiply 4 and -1 to get -4.

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

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

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

Combine -3x and -\frac{1}{3}x to get -\frac{10}{3}x.

$$2x^{2}-\frac{10}{3}x=2\left(2x+1\right)$$

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

$$2x^{2}-\frac{10}{3}x=4x+2$$

Subtract 4x from both sides.

$$2x^{2}-\frac{10}{3}x-4x=2$$

Combine -\frac{10}{3}x and -4x to get -\frac{22}{3}x.

$$2x^{2}-\frac{22}{3}x=2$$

Subtract 2 from both sides.

$$2x^{2}-\frac{22}{3}x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -\frac{22}{3} for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{22}{3}\right)±\sqrt{\left(-\frac{22}{3}\right)^{2}-4\times 2\left(-2\right)}}{2\times 2}$$

Square -\frac{22}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{22}{3}\right)±\sqrt{\frac{484}{9}-4\times 2\left(-2\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-\frac{22}{3}\right)±\sqrt{\frac{484}{9}-8\left(-2\right)}}{2\times 2}$$

Multiply -8 times -2.

$$x=\frac{-\left(-\frac{22}{3}\right)±\sqrt{\frac{484}{9}+16}}{2\times 2}$$

Add \frac{484}{9} to 16.

$$x=\frac{-\left(-\frac{22}{3}\right)±\sqrt{\frac{628}{9}}}{2\times 2}$$

Take the square root of \frac{628}{9}.

$$x=\frac{-\left(-\frac{22}{3}\right)±\frac{2\sqrt{157}}{3}}{2\times 2}$$

The opposite of -\frac{22}{3} is \frac{22}{3}.

$$x=\frac{\frac{22}{3}±\frac{2\sqrt{157}}{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{\frac{22}{3}±\frac{2\sqrt{157}}{3}}{4}$$

Now solve the equation x=\frac{\frac{22}{3}±\frac{2\sqrt{157}}{3}}{4} when ± is plus. Add \frac{22}{3} to \frac{2\sqrt{157}}{3}.

$$x=\frac{2\sqrt{157}+22}{3\times 4}$$

Divide \frac{22+2\sqrt{157}}{3} by 4.

$$x=\frac{\sqrt{157}+11}{6}$$

Now solve the equation x=\frac{\frac{22}{3}±\frac{2\sqrt{157}}{3}}{4} when ± is minus. Subtract \frac{2\sqrt{157}}{3} from \frac{22}{3}.

$$x=\frac{22-2\sqrt{157}}{3\times 4}$$

Divide \frac{22-2\sqrt{157}}{3} by 4.

$$x=\frac{11-\sqrt{157}}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{157}+11}{6} x=\frac{11-\sqrt{157}}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 2,3,4,6.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x}{3} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{3}{3}.

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

Since \frac{2x}{6} and \frac{3}{6} have the same denominator, subtract them by subtracting their numerators.

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

Express 6\times \frac{2x-3}{6} as a single fraction.

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

Cancel out 6 and 6.

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

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

$$2x^{2}-3x+4x\left(\frac{1}{4}-\frac{1}{3}\right)=2\left(2x+1\right)$$

Least common multiple of 4 and 3 is 12. Convert \frac{1}{4} and \frac{1}{3} to fractions with denominator 12.

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

Since \frac{3}{12} and \frac{4}{12} have the same denominator, subtract them by subtracting their numerators.

$$2x^{2}-3x+4x\times \frac{3-4}{12}=2\left(2x+1\right)$$

Subtract 4 from 3 to get -1.

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

Express 4\left(-\frac{1}{12}\right) as a single fraction.

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

Multiply 4 and -1 to get -4.

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

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

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

Combine -3x and -\frac{1}{3}x to get -\frac{10}{3}x.

$$2x^{2}-\frac{10}{3}x=2\left(2x+1\right)$$

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

$$2x^{2}-\frac{10}{3}x=4x+2$$

Subtract 4x from both sides.

$$2x^{2}-\frac{10}{3}x-4x=2$$

Combine -\frac{10}{3}x and -4x to get -\frac{22}{3}x.

$$2x^{2}-\frac{22}{3}x=2$$

Divide both sides by 2.

$$\frac{2x^{2}-\frac{22}{3}x}{2}=\frac{2}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{\frac{22}{3}}{2}\right)x=\frac{2}{2}$$

Divide -\frac{22}{3} by 2.

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

Divide 2 by 2.

$$x^{2}-\frac{11}{3}x=1$$

Divide -\frac{11}{3}, the coefficient of the x term, by 2 to get -\frac{11}{6}. Then add the square of -\frac{11}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{3}x+\left(-\frac{11}{6}\right)^{2}=1+\left(-\frac{11}{6}\right)^{2}$$

Square -\frac{11}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=1+\frac{121}{36}$$

Add 1 to \frac{121}{36}.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=\frac{157}{36}$$

Factor x^{2}-\frac{11}{3}x+\frac{121}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{6}\right)^{2}=\frac{157}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{6}\right)^{2}}=\sqrt{\frac{157}{36}}$$

Simplify.

$$x-\frac{11}{6}=\frac{\sqrt{157}}{6} x-\frac{11}{6}=-\frac{\sqrt{157}}{6}$$

Add \frac{11}{6} to both sides of the equation.

$$x=\frac{\sqrt{157}+11}{6} x=\frac{11-\sqrt{157}}{6}$$