Solve 6x*2x+4=2x+2*2x+12

$$6x \times 2x+4=2x+2 \times 2x+12$$

Solve for x

$$x=\frac{\sqrt{105}}{12}+\frac{1}{4}\approx 1.103912564$$

$$x=-\frac{\sqrt{105}}{12}+\frac{1}{4}\approx -0.603912564$$

Steps Using the Quadratic Formula

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

$$6x^{2}\times 2+4=2x+2\times 2x+12$$

Multiply 6 and 2 to get 12.

$$12x^{2}+4=2x+2\times 2x+12$$

Multiply 2 and 2 to get 4.

$$12x^{2}+4=2x+4x+12$$

Combine 2x and 4x to get 6x.

$$12x^{2}+4=6x+12$$

Subtract 6x from both sides.

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

Subtract 12 from both sides.

$$12x^{2}+4-6x-12=0$$

Subtract 12 from 4 to get -8.

$$12x^{2}-8-6x=0$$

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.

$$12x^{2}-6x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -6 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 12\left(-8\right)}}{2\times 12}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 12\left(-8\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-6\right)±\sqrt{36-48\left(-8\right)}}{2\times 12}$$

Multiply -48 times -8.

$$x=\frac{-\left(-6\right)±\sqrt{36+384}}{2\times 12}$$

Add 36 to 384.

$$x=\frac{-\left(-6\right)±\sqrt{420}}{2\times 12}$$

Take the square root of 420.

$$x=\frac{-\left(-6\right)±2\sqrt{105}}{2\times 12}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{105}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{6±2\sqrt{105}}{24}$$

Now solve the equation x=\frac{6±2\sqrt{105}}{24} when ± is plus. Add 6 to 2\sqrt{105}.

$$x=\frac{2\sqrt{105}+6}{24}$$

Divide 6+2\sqrt{105} by 24.

$$x=\frac{\sqrt{105}}{12}+\frac{1}{4}$$

Now solve the equation x=\frac{6±2\sqrt{105}}{24} when ± is minus. Subtract 2\sqrt{105} from 6.

$$x=\frac{6-2\sqrt{105}}{24}$$

Divide 6-2\sqrt{105} by 24.

$$x=-\frac{\sqrt{105}}{12}+\frac{1}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{105}}{12}+\frac{1}{4} x=-\frac{\sqrt{105}}{12}+\frac{1}{4}$$

Steps for Completing the Square

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

$$6x^{2}\times 2+4=2x+2\times 2x+12$$

Multiply 6 and 2 to get 12.

$$12x^{2}+4=2x+2\times 2x+12$$

Multiply 2 and 2 to get 4.

$$12x^{2}+4=2x+4x+12$$

Combine 2x and 4x to get 6x.

$$12x^{2}+4=6x+12$$

Subtract 6x from both sides.

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

Subtract 4 from both sides.

$$12x^{2}-6x=12-4$$

Subtract 4 from 12 to get 8.

$$12x^{2}-6x=8$$

Divide both sides by 12.

$$\frac{12x^{2}-6x}{12}=\frac{8}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{6}{12}\right)x=\frac{8}{12}$$

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

$$x^{2}-\frac{1}{2}x=\frac{8}{12}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

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

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.

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

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{2}{3}+\frac{1}{16}$$

Add \frac{2}{3} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{35}{48}$$

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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{35}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{35}{48}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{105}}{12} x-\frac{1}{4}=-\frac{\sqrt{105}}{12}$$

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

$$x=\frac{\sqrt{105}}{12}+\frac{1}{4} x=-\frac{\sqrt{105}}{12}+\frac{1}{4}$$