Solve 2x(3x-2)-4=x

$$2x(3x-2)-4=x$$

Solve for x

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

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps Using Factoring By Grouping

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

$$6x^{2}-4x-4=x$$

Subtract x from both sides.

$$6x^{2}-4x-4-x=0$$

Combine -4x and -x to get -5x.

$$6x^{2}-5x-4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-4\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -5.

$$a=-8 b=3$$

Rewrite 6x^{2}-5x-4 as \left(6x^{2}-8x\right)+\left(3x-4\right).

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

Factor out 2x in 6x^{2}-8x.

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

Factor out common term 3x-4 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

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

$$6x^{2}-4x-4=x$$

Subtract x from both sides.

$$6x^{2}-4x-4-x=0$$

Combine -4x and -x to get -5x.

$$6x^{2}-5x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-4\right)}}{2\times 6}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-4\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\left(-4\right)}}{2\times 6}$$

Multiply -24 times -4.

$$x=\frac{-\left(-5\right)±\sqrt{25+96}}{2\times 6}$$

Add 25 to 96.

$$x=\frac{-\left(-5\right)±\sqrt{121}}{2\times 6}$$

Take the square root of 121.

$$x=\frac{-\left(-5\right)±11}{2\times 6}$$

The opposite of -5 is 5.

$$x=\frac{5±11}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{5±11}{12}$$

Now solve the equation x=\frac{5±11}{12} when ± is plus. Add 5 to 11.

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

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

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

Now solve the equation x=\frac{5±11}{12} when ± is minus. Subtract 11 from 5.

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

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

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

The equation is now solved.

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

Steps for Completing the Square

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

$$6x^{2}-4x-4=x$$

Subtract x from both sides.

$$6x^{2}-4x-4-x=0$$

Combine -4x and -x to get -5x.

$$6x^{2}-5x-4=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$6x^{2}-5x=4$$

Divide both sides by 6.

$$\frac{6x^{2}-5x}{6}=\frac{4}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{5}{6}x=\frac{4}{6}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{6}x=\frac{2}{3}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{2}{3}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{2}{3}+\frac{25}{144}$$

Add \frac{2}{3} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{121}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{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}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{121}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{121}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{11}{12} x-\frac{5}{12}=-\frac{11}{12}$$

Add \frac{5}{12} to both sides of the equation.

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