Solve x+1/4x=276-x-1/3x(276-x)

$$x + \frac { 1 } { 4 } x = 276 - x - \frac { 1 } { 3 } x ( 276 - x )$$

Solve for x

$$x = \frac{3 \sqrt{136241} + 1131}{8} \approx 279.790644437$$

$$x = \frac{1131 - 3 \sqrt{136241}}{8} \approx 2.959355563$$

Steps Using the Quadratic Formula

Combine x and \frac{1}{4}x to get \frac{5}{4}x.

$$\frac{5}{4}x=276-x-\frac{1}{3}x\left(276-x\right)$$

Add \frac{1}{3}x\left(276-x\right) to both sides.

$$\frac{5}{4}x+\frac{1}{3}x\left(276-x\right)=276-x$$

Use the distributive property to multiply \frac{1}{3}x by 276-x.

$$\frac{5}{4}x+\frac{1}{3}x\times 276+\frac{1}{3}x\left(-1\right)x=276-x$$

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

$$\frac{5}{4}x+\frac{1}{3}x\times 276+\frac{1}{3}x^{2}\left(-1\right)=276-x$$

Multiply \frac{1}{3} and 276 to get \frac{276}{3}.

$$\frac{5}{4}x+\frac{276}{3}x+\frac{1}{3}x^{2}\left(-1\right)=276-x$$

Divide 276 by 3 to get 92.

$$\frac{5}{4}x+92x+\frac{1}{3}x^{2}\left(-1\right)=276-x$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

$$\frac{5}{4}x+92x-\frac{1}{3}x^{2}=276-x$$

Combine \frac{5}{4}x and 92x to get \frac{373}{4}x.

$$\frac{373}{4}x-\frac{1}{3}x^{2}=276-x$$

Subtract 276 from both sides.

$$\frac{373}{4}x-\frac{1}{3}x^{2}-276=-x$$

Add x to both sides.

$$\frac{373}{4}x-\frac{1}{3}x^{2}-276+x=0$$

Combine \frac{373}{4}x and x to get \frac{377}{4}x.

$$\frac{377}{4}x-\frac{1}{3}x^{2}-276=0$$

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.

$$-\frac{1}{3}x^{2}+\frac{377}{4}x-276=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{3} for a, \frac{377}{4} for b, and -276 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{377}{4}±\sqrt{\left(\frac{377}{4}\right)^{2}-4\left(-\frac{1}{3}\right)\left(-276\right)}}{2\left(-\frac{1}{3}\right)}$$

Square \frac{377}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{377}{4}±\sqrt{\frac{142129}{16}-4\left(-\frac{1}{3}\right)\left(-276\right)}}{2\left(-\frac{1}{3}\right)}$$

Multiply -4 times -\frac{1}{3}.

$$x=\frac{-\frac{377}{4}±\sqrt{\frac{142129}{16}+\frac{4}{3}\left(-276\right)}}{2\left(-\frac{1}{3}\right)}$$

Multiply \frac{4}{3} times -276.

$$x=\frac{-\frac{377}{4}±\sqrt{\frac{142129}{16}-368}}{2\left(-\frac{1}{3}\right)}$$

Add \frac{142129}{16} to -368.

$$x=\frac{-\frac{377}{4}±\sqrt{\frac{136241}{16}}}{2\left(-\frac{1}{3}\right)}$$

Take the square root of \frac{136241}{16}.

$$x=\frac{-\frac{377}{4}±\frac{\sqrt{136241}}{4}}{2\left(-\frac{1}{3}\right)}$$

Multiply 2 times -\frac{1}{3}.

$$x=\frac{-\frac{377}{4}±\frac{\sqrt{136241}}{4}}{-\frac{2}{3}}$$

Now solve the equation x=\frac{-\frac{377}{4}±\frac{\sqrt{136241}}{4}}{-\frac{2}{3}} when ± is plus. Add -\frac{377}{4} to \frac{\sqrt{136241}}{4}.

$$x=\frac{\sqrt{136241}-377}{-\frac{2}{3}\times 4}$$

Divide \frac{-377+\sqrt{136241}}{4} by -\frac{2}{3} by multiplying \frac{-377+\sqrt{136241}}{4} by the reciprocal of -\frac{2}{3}.

$$x=\frac{1131-3\sqrt{136241}}{8}$$

Now solve the equation x=\frac{-\frac{377}{4}±\frac{\sqrt{136241}}{4}}{-\frac{2}{3}} when ± is minus. Subtract \frac{\sqrt{136241}}{4} from -\frac{377}{4}.

$$x=\frac{-\sqrt{136241}-377}{-\frac{2}{3}\times 4}$$

Divide \frac{-377-\sqrt{136241}}{4} by -\frac{2}{3} by multiplying \frac{-377-\sqrt{136241}}{4} by the reciprocal of -\frac{2}{3}.

$$x=\frac{3\sqrt{136241}+1131}{8}$$

The equation is now solved.

$$x=\frac{1131-3\sqrt{136241}}{8} x=\frac{3\sqrt{136241}+1131}{8}$$

Steps for Completing the Square

Combine x and \frac{1}{4}x to get \frac{5}{4}x.

$$\frac{5}{4}x=276-x-\frac{1}{3}x\left(276-x\right)$$

Add \frac{1}{3}x\left(276-x\right) to both sides.

$$\frac{5}{4}x+\frac{1}{3}x\left(276-x\right)=276-x$$

Use the distributive property to multiply \frac{1}{3}x by 276-x.

$$\frac{5}{4}x+\frac{1}{3}x\times 276+\frac{1}{3}x\left(-1\right)x=276-x$$

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

$$\frac{5}{4}x+\frac{1}{3}x\times 276+\frac{1}{3}x^{2}\left(-1\right)=276-x$$

Multiply \frac{1}{3} and 276 to get \frac{276}{3}.

$$\frac{5}{4}x+\frac{276}{3}x+\frac{1}{3}x^{2}\left(-1\right)=276-x$$

Divide 276 by 3 to get 92.

$$\frac{5}{4}x+92x+\frac{1}{3}x^{2}\left(-1\right)=276-x$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

$$\frac{5}{4}x+92x-\frac{1}{3}x^{2}=276-x$$

Combine \frac{5}{4}x and 92x to get \frac{373}{4}x.

$$\frac{373}{4}x-\frac{1}{3}x^{2}=276-x$$

Add x to both sides.

$$\frac{373}{4}x-\frac{1}{3}x^{2}+x=276$$

Combine \frac{373}{4}x and x to get \frac{377}{4}x.

$$\frac{377}{4}x-\frac{1}{3}x^{2}=276$$

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.

$$-\frac{1}{3}x^{2}+\frac{377}{4}x=276$$

Multiply both sides by -3.

$$\frac{-\frac{1}{3}x^{2}+\frac{377}{4}x}{-\frac{1}{3}}=\frac{276}{-\frac{1}{3}}$$

Dividing by -\frac{1}{3} undoes the multiplication by -\frac{1}{3}.

$$x^{2}+\frac{\frac{377}{4}}{-\frac{1}{3}}x=\frac{276}{-\frac{1}{3}}$$

Divide \frac{377}{4} by -\frac{1}{3} by multiplying \frac{377}{4} by the reciprocal of -\frac{1}{3}.

$$x^{2}-\frac{1131}{4}x=\frac{276}{-\frac{1}{3}}$$

Divide 276 by -\frac{1}{3} by multiplying 276 by the reciprocal of -\frac{1}{3}.

$$x^{2}-\frac{1131}{4}x=-828$$

Divide -\frac{1131}{4}, the coefficient of the x term, by 2 to get -\frac{1131}{8}. Then add the square of -\frac{1131}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1131}{4}x+\left(-\frac{1131}{8}\right)^{2}=-828+\left(-\frac{1131}{8}\right)^{2}$$

Square -\frac{1131}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1131}{4}x+\frac{1279161}{64}=-828+\frac{1279161}{64}$$

Add -828 to \frac{1279161}{64}.

$$x^{2}-\frac{1131}{4}x+\frac{1279161}{64}=\frac{1226169}{64}$$

Factor x^{2}-\frac{1131}{4}x+\frac{1279161}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1131}{8}\right)^{2}=\frac{1226169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1131}{8}\right)^{2}}=\sqrt{\frac{1226169}{64}}$$

Simplify.

$$x-\frac{1131}{8}=\frac{3\sqrt{136241}}{8} x-\frac{1131}{8}=-\frac{3\sqrt{136241}}{8}$$

Add \frac{1131}{8} to both sides of the equation.

$$x=\frac{3\sqrt{136241}+1131}{8} x=\frac{1131-3\sqrt{136241}}{8}$$