Solve 12x=-12x^2+7

$$12 x = - 12 x ^ { 2 } + 7$$

Solve for x

$$x=\frac{\sqrt{30}}{6}-\frac{1}{2}\approx 0.412870929$$

$$x=-\frac{\sqrt{30}}{6}-\frac{1}{2}\approx -1.412870929$$

Steps Using the Quadratic Formula

Add 12x^{2} to both sides.

$$12x+12x^{2}=7$$

Subtract 7 from both sides.

$$12x+12x^{2}-7=0$$

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.

$$12x^{2}+12x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 12 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 12\left(-7\right)}}{2\times 12}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 12\left(-7\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-12±\sqrt{144-48\left(-7\right)}}{2\times 12}$$

Multiply -48 times -7.

$$x=\frac{-12±\sqrt{144+336}}{2\times 12}$$

Add 144 to 336.

$$x=\frac{-12±\sqrt{480}}{2\times 12}$$

Take the square root of 480.

$$x=\frac{-12±4\sqrt{30}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-12±4\sqrt{30}}{24}$$

Now solve the equation x=\frac{-12±4\sqrt{30}}{24} when ± is plus. Add -12 to 4\sqrt{30}.

$$x=\frac{4\sqrt{30}-12}{24}$$

Divide -12+4\sqrt{30} by 24.

$$x=\frac{\sqrt{30}}{6}-\frac{1}{2}$$

Now solve the equation x=\frac{-12±4\sqrt{30}}{24} when ± is minus. Subtract 4\sqrt{30} from -12.

$$x=\frac{-4\sqrt{30}-12}{24}$$

Divide -12-4\sqrt{30} by 24.

$$x=-\frac{\sqrt{30}}{6}-\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{30}}{6}-\frac{1}{2} x=-\frac{\sqrt{30}}{6}-\frac{1}{2}$$

Steps for Completing the Square

Add 12x^{2} to both sides.

$$12x+12x^{2}=7$$

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.

$$12x^{2}+12x=7$$

Divide both sides by 12.

$$\frac{12x^{2}+12x}{12}=\frac{7}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{12}{12}x=\frac{7}{12}$$

Divide 12 by 12.

$$x^{2}+x=\frac{7}{12}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{7}{12}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{7}{12}+\frac{1}{4}$$

Add \frac{7}{12} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{5}{6}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{5}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{6}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{30}}{6} x+\frac{1}{2}=-\frac{\sqrt{30}}{6}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{30}}{6}-\frac{1}{2} x=-\frac{\sqrt{30}}{6}-\frac{1}{2}$$