Solve 2a/a-a/6=a/3-5/3

$$\frac { 2 a } { a } - \frac { a } { 6 } = \frac { a } { 3 } - \frac { 5 } { 3 }$$

Solve for a

$$a = \frac{22}{3} = 7\frac{1}{3} \approx 7.333333333$$

Steps Using Factoring

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6a, the least common multiple of a,6,3.

$$6\times 2a-aa=2aa+6a\left(-\frac{5}{3}\right)$$

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

$$6\times 2a-a^{2}=2aa+6a\left(-\frac{5}{3}\right)$$

Multiply 6 and 2 to get 12.

$$12a-a^{2}=2aa+6a\left(-\frac{5}{3}\right)$$

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

$$12a-a^{2}=2a^{2}+6a\left(-\frac{5}{3}\right)$$

Express 6\left(-\frac{5}{3}\right) as a single fraction.

$$12a-a^{2}=2a^{2}+\frac{6\left(-5\right)}{3}a$$

Multiply 6 and -5 to get -30.

$$12a-a^{2}=2a^{2}+\frac{-30}{3}a$$

Divide -30 by 3 to get -10.

$$12a-a^{2}=2a^{2}-10a$$

Subtract 2a^{2} from both sides.

$$12a-a^{2}-2a^{2}=-10a$$

Add 10a to both sides.

$$12a-a^{2}-2a^{2}+10a=0$$

Combine 12a and 10a to get 22a.

$$22a-a^{2}-2a^{2}=0$$

Combine -a^{2} and -2a^{2} to get -3a^{2}.

$$22a-3a^{2}=0$$

Factor out a.

$$a\left(22-3a\right)=0$$

To find equation solutions, solve a=0 and 22-3a=0.

$$a=0 a=\frac{22}{3}$$

Variable a cannot be equal to 0.

$$a=\frac{22}{3}$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6a, the least common multiple of a,6,3.

$$6\times 2a-aa=2aa+6a\left(-\frac{5}{3}\right)$$

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

$$6\times 2a-a^{2}=2aa+6a\left(-\frac{5}{3}\right)$$

Multiply 6 and 2 to get 12.

$$12a-a^{2}=2aa+6a\left(-\frac{5}{3}\right)$$

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

$$12a-a^{2}=2a^{2}+6a\left(-\frac{5}{3}\right)$$

Express 6\left(-\frac{5}{3}\right) as a single fraction.

$$12a-a^{2}=2a^{2}+\frac{6\left(-5\right)}{3}a$$

Multiply 6 and -5 to get -30.

$$12a-a^{2}=2a^{2}+\frac{-30}{3}a$$

Divide -30 by 3 to get -10.

$$12a-a^{2}=2a^{2}-10a$$

Subtract 2a^{2} from both sides.

$$12a-a^{2}-2a^{2}=-10a$$

Add 10a to both sides.

$$12a-a^{2}-2a^{2}+10a=0$$

Combine 12a and 10a to get 22a.

$$22a-a^{2}-2a^{2}=0$$

Combine -a^{2} and -2a^{2} to get -3a^{2}.

$$22a-3a^{2}=0$$

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.

$$-3a^{2}+22a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 22 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-22±\sqrt{22^{2}}}{2\left(-3\right)}$$

Take the square root of 22^{2}.

$$a=\frac{-22±22}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a=\frac{-22±22}{-6}$$

Now solve the equation a=\frac{-22±22}{-6} when ± is plus. Add -22 to 22.

$$a=\frac{0}{-6}$$

Divide 0 by -6.

$$a=0$$

Now solve the equation a=\frac{-22±22}{-6} when ± is minus. Subtract 22 from -22.

$$a=-\frac{44}{-6}$$

Reduce the fraction \frac{-44}{-6} to lowest terms by extracting and canceling out 2.

$$a=\frac{22}{3}$$

The equation is now solved.

$$a=0 a=\frac{22}{3}$$

Variable a cannot be equal to 0.

$$a=\frac{22}{3}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6a, the least common multiple of a,6,3.

$$6\times 2a-aa=2aa+6a\left(-\frac{5}{3}\right)$$

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

$$6\times 2a-a^{2}=2aa+6a\left(-\frac{5}{3}\right)$$

Multiply 6 and 2 to get 12.

$$12a-a^{2}=2aa+6a\left(-\frac{5}{3}\right)$$

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

$$12a-a^{2}=2a^{2}+6a\left(-\frac{5}{3}\right)$$

Express 6\left(-\frac{5}{3}\right) as a single fraction.

$$12a-a^{2}=2a^{2}+\frac{6\left(-5\right)}{3}a$$

Multiply 6 and -5 to get -30.

$$12a-a^{2}=2a^{2}+\frac{-30}{3}a$$

Divide -30 by 3 to get -10.

$$12a-a^{2}=2a^{2}-10a$$

Subtract 2a^{2} from both sides.

$$12a-a^{2}-2a^{2}=-10a$$

Add 10a to both sides.

$$12a-a^{2}-2a^{2}+10a=0$$

Combine 12a and 10a to get 22a.

$$22a-a^{2}-2a^{2}=0$$

Combine -a^{2} and -2a^{2} to get -3a^{2}.

$$22a-3a^{2}=0$$

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.

$$-3a^{2}+22a=0$$

Divide both sides by -3.

$$\frac{-3a^{2}+22a}{-3}=\frac{0}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$a^{2}+\frac{22}{-3}a=\frac{0}{-3}$$

Divide 22 by -3.

$$a^{2}-\frac{22}{3}a=\frac{0}{-3}$$

Divide 0 by -3.

$$a^{2}-\frac{22}{3}a=0$$

Divide -\frac{22}{3}, the coefficient of the x term, by 2 to get -\frac{11}{3}. Then add the square of -\frac{11}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{22}{3}a+\left(-\frac{11}{3}\right)^{2}=\left(-\frac{11}{3}\right)^{2}$$

Square -\frac{11}{3} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{22}{3}a+\frac{121}{9}=\frac{121}{9}$$

Factor a^{2}-\frac{22}{3}a+\frac{121}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{11}{3}\right)^{2}=\frac{121}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{11}{3}\right)^{2}}=\sqrt{\frac{121}{9}}$$

Simplify.

$$a-\frac{11}{3}=\frac{11}{3} a-\frac{11}{3}=-\frac{11}{3}$$

Add \frac{11}{3} to both sides of the equation.

$$a=\frac{22}{3} a=0$$

Variable a cannot be equal to 0.

$$a=\frac{22}{3}$$