Solve x/4+3/4xdiv1/3x-3/4xdivfrac{1}{6}x+30=x

$$\frac{ x }{ 4 } + \frac{ 3 }{ 4 } x \div \frac{ 1 }{ 3 } x- \frac{ 3 }{ 4 } x \div \frac{ 1 }{ 6 } x+30=x$$

Solve for x

$$x = \frac{\sqrt{481} - 1}{6} \approx 3.4886187$$

$$x=\frac{-\sqrt{481}-1}{6}\approx -3.821952033$$

Steps Using the Quadratic Formula

Divide \frac{3}{4}x by \frac{1}{3} to get \frac{9}{4}x.

$$\frac{x}{4}+\frac{9}{4}xx-\frac{\frac{3}{4}x}{\frac{1}{6}}x+30=x$$

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

$$\frac{x}{4}+\frac{9}{4}x^{2}-\frac{\frac{3}{4}x}{\frac{1}{6}}x+30=x$$

Divide \frac{3}{4}x by \frac{1}{6} to get \frac{9}{2}x.

$$\frac{x}{4}+\frac{9}{4}x^{2}-\frac{9}{2}xx+30=x$$

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

$$\frac{x}{4}+\frac{9}{4}x^{2}-\frac{9}{2}x^{2}+30=x$$

Combine \frac{9}{4}x^{2} and -\frac{9}{2}x^{2} to get -\frac{9}{4}x^{2}.

$$\frac{x}{4}-\frac{9}{4}x^{2}+30=x$$

Subtract x from both sides.

$$\frac{x}{4}-\frac{9}{4}x^{2}+30-x=0$$

Combine \frac{x}{4} and -x to get -\frac{3}{4}x.

$$-\frac{3}{4}x-\frac{9}{4}x^{2}+30=0$$

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.

$$-\frac{9}{4}x^{2}-\frac{3}{4}x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{9}{4} for a, -\frac{3}{4} for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\left(-\frac{3}{4}\right)^{2}-4\left(-\frac{9}{4}\right)\times 30}}{2\left(-\frac{9}{4}\right)}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-4\left(-\frac{9}{4}\right)\times 30}}{2\left(-\frac{9}{4}\right)}$$

Multiply -4 times -\frac{9}{4}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+9\times 30}}{2\left(-\frac{9}{4}\right)}$$

Multiply 9 times 30.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+270}}{2\left(-\frac{9}{4}\right)}$$

Add \frac{9}{16} to 270.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{4329}{16}}}{2\left(-\frac{9}{4}\right)}$$

Take the square root of \frac{4329}{16}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\frac{3\sqrt{481}}{4}}{2\left(-\frac{9}{4}\right)}$$

The opposite of -\frac{3}{4} is \frac{3}{4}.

$$x=\frac{\frac{3}{4}±\frac{3\sqrt{481}}{4}}{2\left(-\frac{9}{4}\right)}$$

Multiply 2 times -\frac{9}{4}.

$$x=\frac{\frac{3}{4}±\frac{3\sqrt{481}}{4}}{-\frac{9}{2}}$$

Now solve the equation x=\frac{\frac{3}{4}±\frac{3\sqrt{481}}{4}}{-\frac{9}{2}} when ± is plus. Add \frac{3}{4} to \frac{3\sqrt{481}}{4}.

$$x=\frac{3\sqrt{481}+3}{-\frac{9}{2}\times 4}$$

Divide \frac{3+3\sqrt{481}}{4} by -\frac{9}{2} by multiplying \frac{3+3\sqrt{481}}{4} by the reciprocal of -\frac{9}{2}.

$$x=\frac{-\sqrt{481}-1}{6}$$

Now solve the equation x=\frac{\frac{3}{4}±\frac{3\sqrt{481}}{4}}{-\frac{9}{2}} when ± is minus. Subtract \frac{3\sqrt{481}}{4} from \frac{3}{4}.

$$x=\frac{3-3\sqrt{481}}{-\frac{9}{2}\times 4}$$

Divide \frac{3-3\sqrt{481}}{4} by -\frac{9}{2} by multiplying \frac{3-3\sqrt{481}}{4} by the reciprocal of -\frac{9}{2}.

$$x=\frac{\sqrt{481}-1}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{481}-1}{6} x=\frac{\sqrt{481}-1}{6}$$

Steps for Completing the Square

Divide \frac{3}{4}x by \frac{1}{3} to get \frac{9}{4}x.

$$\frac{x}{4}+\frac{9}{4}xx-\frac{\frac{3}{4}x}{\frac{1}{6}}x+30=x$$

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

$$\frac{x}{4}+\frac{9}{4}x^{2}-\frac{\frac{3}{4}x}{\frac{1}{6}}x+30=x$$

Divide \frac{3}{4}x by \frac{1}{6} to get \frac{9}{2}x.

$$\frac{x}{4}+\frac{9}{4}x^{2}-\frac{9}{2}xx+30=x$$

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

$$\frac{x}{4}+\frac{9}{4}x^{2}-\frac{9}{2}x^{2}+30=x$$

Combine \frac{9}{4}x^{2} and -\frac{9}{2}x^{2} to get -\frac{9}{4}x^{2}.

$$\frac{x}{4}-\frac{9}{4}x^{2}+30=x$$

Subtract x from both sides.

$$\frac{x}{4}-\frac{9}{4}x^{2}+30-x=0$$

Combine \frac{x}{4} and -x to get -\frac{3}{4}x.

$$-\frac{3}{4}x-\frac{9}{4}x^{2}+30=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{3}{4}x-\frac{9}{4}x^{2}=-30$$

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.

$$-\frac{9}{4}x^{2}-\frac{3}{4}x=-30$$

Divide both sides of the equation by -\frac{9}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{9}{4}x^{2}-\frac{3}{4}x}{-\frac{9}{4}}=-\frac{30}{-\frac{9}{4}}$$

Dividing by -\frac{9}{4} undoes the multiplication by -\frac{9}{4}.

$$x^{2}+\left(-\frac{\frac{3}{4}}{-\frac{9}{4}}\right)x=-\frac{30}{-\frac{9}{4}}$$

Divide -\frac{3}{4} by -\frac{9}{4} by multiplying -\frac{3}{4} by the reciprocal of -\frac{9}{4}.

$$x^{2}+\frac{1}{3}x=-\frac{30}{-\frac{9}{4}}$$

Divide -30 by -\frac{9}{4} by multiplying -30 by the reciprocal of -\frac{9}{4}.

$$x^{2}+\frac{1}{3}x=\frac{40}{3}$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=\frac{40}{3}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{40}{3}+\frac{1}{36}$$

Add \frac{40}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{481}{36}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=\frac{481}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{\frac{481}{36}}$$

Simplify.

$$x+\frac{1}{6}=\frac{\sqrt{481}}{6} x+\frac{1}{6}=-\frac{\sqrt{481}}{6}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{\sqrt{481}-1}{6} x=\frac{-\sqrt{481}-1}{6}$$