Solve x/2+3=3x*(x+2)

$$\frac{ x }{ 2 } +3=3x \times (x+2)$$

Solve for x

$$x=\frac{\sqrt{265}-11}{12}\approx 0.439901716$$

$$x=\frac{-\sqrt{265}-11}{12}\approx -2.27323505$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x+6=6x\left(x+2\right)$$

Use the distributive property to multiply 6x by x+2.

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

Subtract 6x^{2} from both sides.

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

Subtract 12x from both sides.

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

Combine x and -12x to get -11x.

$$-11x+6-6x^{2}=0$$

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.

$$-6x^{2}-11x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -11 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-6\right)\times 6}}{2\left(-6\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-6\right)\times 6}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-11\right)±\sqrt{121+24\times 6}}{2\left(-6\right)}$$

Multiply 24 times 6.

$$x=\frac{-\left(-11\right)±\sqrt{121+144}}{2\left(-6\right)}$$

Add 121 to 144.

$$x=\frac{-\left(-11\right)±\sqrt{265}}{2\left(-6\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{265}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{11±\sqrt{265}}{-12}$$

Now solve the equation x=\frac{11±\sqrt{265}}{-12} when ± is plus. Add 11 to \sqrt{265}.

$$x=\frac{\sqrt{265}+11}{-12}$$

Divide 11+\sqrt{265} by -12.

$$x=\frac{-\sqrt{265}-11}{12}$$

Now solve the equation x=\frac{11±\sqrt{265}}{-12} when ± is minus. Subtract \sqrt{265} from 11.

$$x=\frac{11-\sqrt{265}}{-12}$$

Divide 11-\sqrt{265} by -12.

$$x=\frac{\sqrt{265}-11}{12}$$

The equation is now solved.

$$x=\frac{-\sqrt{265}-11}{12} x=\frac{\sqrt{265}-11}{12}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x+6=6x\left(x+2\right)$$

Use the distributive property to multiply 6x by x+2.

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

Subtract 6x^{2} from both sides.

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

Subtract 12x from both sides.

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

Combine x and -12x to get -11x.

$$-11x+6-6x^{2}=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-11x-6x^{2}=-6$$

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.

$$-6x^{2}-11x=-6$$

Divide both sides by -6.

$$\frac{-6x^{2}-11x}{-6}=-\frac{6}{-6}$$

Dividing by -6 undoes the multiplication by -6.

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

Divide -11 by -6.

$$x^{2}+\frac{11}{6}x=-\frac{6}{-6}$$

Divide -6 by -6.

$$x^{2}+\frac{11}{6}x=1$$

Divide \frac{11}{6}, the coefficient of the x term, by 2 to get \frac{11}{12}. Then add the square of \frac{11}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{6}x+\left(\frac{11}{12}\right)^{2}=1+\left(\frac{11}{12}\right)^{2}$$

Square \frac{11}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=1+\frac{121}{144}$$

Add 1 to \frac{121}{144}.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{265}{144}$$

Factor x^{2}+\frac{11}{6}x+\frac{121}{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}.

$$\left(x+\frac{11}{12}\right)^{2}=\frac{265}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{12}\right)^{2}}=\sqrt{\frac{265}{144}}$$

Simplify.

$$x+\frac{11}{12}=\frac{\sqrt{265}}{12} x+\frac{11}{12}=-\frac{\sqrt{265}}{12}$$

Subtract \frac{11}{12} from both sides of the equation.

$$x=\frac{\sqrt{265}-11}{12} x=\frac{-\sqrt{265}-11}{12}$$