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

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

Solve for x

$$x = \frac{\sqrt{129} + 13}{8} \approx 3.044727086$$

$$x=\frac{13-\sqrt{129}}{8}\approx 0.205272914$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply 3 by 2x+1.

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

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

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

Combine 6x and 4x to get 10x.

$$10x+3-2=6\left(2x-1\right)\left(x-\left(1+\frac{1}{3}x\right)\right)$$

Subtract 2 from 3 to get 1.

$$10x+1=6\left(2x-1\right)\left(x-\left(1+\frac{1}{3}x\right)\right)$$

To find the opposite of 1+\frac{1}{3}x, find the opposite of each term.

$$10x+1=6\left(2x-1\right)\left(x-1-\frac{1}{3}x\right)$$

Combine x and -\frac{1}{3}x to get \frac{2}{3}x.

$$10x+1=6\left(2x-1\right)\left(\frac{2}{3}x-1\right)$$

Use the distributive property to multiply 6 by 2x-1.

$$10x+1=\left(12x-6\right)\left(\frac{2}{3}x-1\right)$$

Apply the distributive property by multiplying each term of 12x-6 by each term of \frac{2}{3}x-1.

$$10x+1=12x\times \frac{2}{3}x-12x-6\times \frac{2}{3}x+6$$

Multiply x and x to get x^{2}.

$$10x+1=12x^{2}\times \frac{2}{3}-12x-6\times \frac{2}{3}x+6$$

Express 12\times \frac{2}{3} as a single fraction.

$$10x+1=\frac{12\times 2}{3}x^{2}-12x-6\times \frac{2}{3}x+6$$

Multiply 12 and 2 to get 24.

$$10x+1=\frac{24}{3}x^{2}-12x-6\times \frac{2}{3}x+6$$

Divide 24 by 3 to get 8.

$$10x+1=8x^{2}-12x-6\times \frac{2}{3}x+6$$

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

$$10x+1=8x^{2}-12x+\frac{-6\times 2}{3}x+6$$

Multiply -6 and 2 to get -12.

$$10x+1=8x^{2}-12x+\frac{-12}{3}x+6$$

Divide -12 by 3 to get -4.

$$10x+1=8x^{2}-12x-4x+6$$

Combine -12x and -4x to get -16x.

$$10x+1=8x^{2}-16x+6$$

Subtract 8x^{2} from both sides.

$$10x+1-8x^{2}=-16x+6$$

Add 16x to both sides.

$$10x+1-8x^{2}+16x=6$$

Combine 10x and 16x to get 26x.

$$26x+1-8x^{2}=6$$

Subtract 6 from both sides.

$$26x+1-8x^{2}-6=0$$

Subtract 6 from 1 to get -5.

$$26x-5-8x^{2}=0$$

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.

$$-8x^{2}+26x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 26 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26±\sqrt{26^{2}-4\left(-8\right)\left(-5\right)}}{2\left(-8\right)}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\left(-8\right)\left(-5\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-26±\sqrt{676+32\left(-5\right)}}{2\left(-8\right)}$$

Multiply 32 times -5.

$$x=\frac{-26±\sqrt{676-160}}{2\left(-8\right)}$$

Add 676 to -160.

$$x=\frac{-26±\sqrt{516}}{2\left(-8\right)}$$

Take the square root of 516.

$$x=\frac{-26±2\sqrt{129}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-26±2\sqrt{129}}{-16}$$

Now solve the equation x=\frac{-26±2\sqrt{129}}{-16} when ± is plus. Add -26 to 2\sqrt{129}.

$$x=\frac{2\sqrt{129}-26}{-16}$$

Divide -26+2\sqrt{129} by -16.

$$x=\frac{13-\sqrt{129}}{8}$$

Now solve the equation x=\frac{-26±2\sqrt{129}}{-16} when ± is minus. Subtract 2\sqrt{129} from -26.

$$x=\frac{-2\sqrt{129}-26}{-16}$$

Divide -26-2\sqrt{129} by -16.

$$x=\frac{\sqrt{129}+13}{8}$$

The equation is now solved.

$$x=\frac{13-\sqrt{129}}{8} x=\frac{\sqrt{129}+13}{8}$$

Steps for Completing the Square

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

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

Use the distributive property to multiply 3 by 2x+1.

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

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

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

Combine 6x and 4x to get 10x.

$$10x+3-2=6\left(2x-1\right)\left(x-\left(1+\frac{1}{3}x\right)\right)$$

Subtract 2 from 3 to get 1.

$$10x+1=6\left(2x-1\right)\left(x-\left(1+\frac{1}{3}x\right)\right)$$

To find the opposite of 1+\frac{1}{3}x, find the opposite of each term.

$$10x+1=6\left(2x-1\right)\left(x-1-\frac{1}{3}x\right)$$

Combine x and -\frac{1}{3}x to get \frac{2}{3}x.

$$10x+1=6\left(2x-1\right)\left(\frac{2}{3}x-1\right)$$

Use the distributive property to multiply 6 by 2x-1.

$$10x+1=\left(12x-6\right)\left(\frac{2}{3}x-1\right)$$

Apply the distributive property by multiplying each term of 12x-6 by each term of \frac{2}{3}x-1.

$$10x+1=12x\times \frac{2}{3}x-12x-6\times \frac{2}{3}x+6$$

Multiply x and x to get x^{2}.

$$10x+1=12x^{2}\times \frac{2}{3}-12x-6\times \frac{2}{3}x+6$$

Express 12\times \frac{2}{3} as a single fraction.

$$10x+1=\frac{12\times 2}{3}x^{2}-12x-6\times \frac{2}{3}x+6$$

Multiply 12 and 2 to get 24.

$$10x+1=\frac{24}{3}x^{2}-12x-6\times \frac{2}{3}x+6$$

Divide 24 by 3 to get 8.

$$10x+1=8x^{2}-12x-6\times \frac{2}{3}x+6$$

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

$$10x+1=8x^{2}-12x+\frac{-6\times 2}{3}x+6$$

Multiply -6 and 2 to get -12.

$$10x+1=8x^{2}-12x+\frac{-12}{3}x+6$$

Divide -12 by 3 to get -4.

$$10x+1=8x^{2}-12x-4x+6$$

Combine -12x and -4x to get -16x.

$$10x+1=8x^{2}-16x+6$$

Subtract 8x^{2} from both sides.

$$10x+1-8x^{2}=-16x+6$$

Add 16x to both sides.

$$10x+1-8x^{2}+16x=6$$

Combine 10x and 16x to get 26x.

$$26x+1-8x^{2}=6$$

Subtract 1 from both sides.

$$26x-8x^{2}=6-1$$

Subtract 1 from 6 to get 5.

$$26x-8x^{2}=5$$

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.

$$-8x^{2}+26x=5$$

Divide both sides by -8.

$$\frac{-8x^{2}+26x}{-8}=\frac{5}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{26}{-8}x=\frac{5}{-8}$$

Reduce the fraction \frac{26}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{13}{4}x=\frac{5}{-8}$$

Divide 5 by -8.

$$x^{2}-\frac{13}{4}x=-\frac{5}{8}$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=-\frac{5}{8}+\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=-\frac{5}{8}+\frac{169}{64}$$

Add -\frac{5}{8} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{129}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{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}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{129}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{129}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{\sqrt{129}}{8} x-\frac{13}{8}=-\frac{\sqrt{129}}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=\frac{\sqrt{129}+13}{8} x=\frac{13-\sqrt{129}}{8}$$