Solve (2x-1)(2x+1)/3=3x-2/6+x^2/3

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

Solve for x

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

$$x=0$$

Steps Using Factoring

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

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

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

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

Use the distributive property to multiply 4x-2 by 2x+1 and combine like terms.

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

Subtract 3x from both sides.

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

Subtract -2 from both sides.

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

The opposite of -2 is 2.

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

Subtract 2x^{2} from both sides.

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

Add -2 and 2 to get 0.

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

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

$$6x^{2}-3x=0$$

Factor out x.

$$x\left(6x-3\right)=0$$

To find equation solutions, solve x=0 and 6x-3=0.

$$x=0 x=\frac{1}{2}$$

Steps Using the Quadratic Formula

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

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

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

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

Use the distributive property to multiply 4x-2 by 2x+1 and combine like terms.

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

Subtract 3x from both sides.

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

Subtract -2 from both sides.

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

The opposite of -2 is 2.

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

Subtract 2x^{2} from both sides.

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

Add -2 and 2 to get 0.

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

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

$$6x^{2}-3x=0$$

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

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}}}{2\times 6}$$

Take the square root of \left(-3\right)^{2}.

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

The opposite of -3 is 3.

$$x=\frac{3±3}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{3±3}{12}$$

Now solve the equation x=\frac{3±3}{12} when ± is plus. Add 3 to 3.

$$x=\frac{6}{12}$$

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

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

Now solve the equation x=\frac{3±3}{12} when ± is minus. Subtract 3 from 3.

$$x=\frac{0}{12}$$

Divide 0 by 12.

$$x=0$$

The equation is now solved.

$$x=\frac{1}{2} x=0$$

Steps for Completing the Square

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

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

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

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

Use the distributive property to multiply 4x-2 by 2x+1 and combine like terms.

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

Subtract 3x from both sides.

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

Subtract 2x^{2} from both sides.

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

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

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

Add 2 to both sides.

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

Add -2 and 2 to get 0.

$$6x^{2}-3x=0$$

Divide both sides by 6.

$$\frac{6x^{2}-3x}{6}=\frac{0}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{3}{6}\right)x=\frac{0}{6}$$

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

$$x^{2}-\frac{1}{2}x=\frac{0}{6}$$

Divide 0 by 6.

$$x^{2}-\frac{1}{2}x=0$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

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

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1}{2} x=0$$