Solve 6(-18/a+a/2)=20

$$6 ( - \frac { 18 } { a } + \frac { a } { 2 } ) = 20$$

Solve for a

$$a = \frac{2 \sqrt{106} + 10}{3} \approx 10.197086761$$

$$a=\frac{10-2\sqrt{106}}{3}\approx -3.530420094$$

Steps Using the Quadratic Formula

Divide both sides by 6.

$$-\frac{18}{a}+\frac{a}{2}=\frac{20}{6}$$

Reduce the fraction \frac{20}{6} to lowest terms by extracting and canceling out 2.

$$-\frac{18}{a}+\frac{a}{2}=\frac{10}{3}$$

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,2,3.

$$-6\times 18+3aa=20a$$

Multiply -6 and 18 to get -108.

$$-108+3aa=20a$$

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

$$-108+3a^{2}=20a$$

Subtract 20a from both sides.

$$-108+3a^{2}-20a=0$$

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.

$$3a^{2}-20a-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -20 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 3\left(-108\right)}}{2\times 3}$$

Square -20.

$$a=\frac{-\left(-20\right)±\sqrt{400-4\times 3\left(-108\right)}}{2\times 3}$$

Multiply -4 times 3.

$$a=\frac{-\left(-20\right)±\sqrt{400-12\left(-108\right)}}{2\times 3}$$

Multiply -12 times -108.

$$a=\frac{-\left(-20\right)±\sqrt{400+1296}}{2\times 3}$$

Add 400 to 1296.

$$a=\frac{-\left(-20\right)±\sqrt{1696}}{2\times 3}$$

Take the square root of 1696.

$$a=\frac{-\left(-20\right)±4\sqrt{106}}{2\times 3}$$

The opposite of -20 is 20.

$$a=\frac{20±4\sqrt{106}}{2\times 3}$$

Multiply 2 times 3.

$$a=\frac{20±4\sqrt{106}}{6}$$

Now solve the equation a=\frac{20±4\sqrt{106}}{6} when ± is plus. Add 20 to 4\sqrt{106}.

$$a=\frac{4\sqrt{106}+20}{6}$$

Divide 20+4\sqrt{106} by 6.

$$a=\frac{2\sqrt{106}+10}{3}$$

Now solve the equation a=\frac{20±4\sqrt{106}}{6} when ± is minus. Subtract 4\sqrt{106} from 20.

$$a=\frac{20-4\sqrt{106}}{6}$$

Divide 20-4\sqrt{106} by 6.

$$a=\frac{10-2\sqrt{106}}{3}$$

The equation is now solved.

$$a=\frac{2\sqrt{106}+10}{3} a=\frac{10-2\sqrt{106}}{3}$$

Steps for Completing the Square

Divide both sides by 6.

$$-\frac{18}{a}+\frac{a}{2}=\frac{20}{6}$$

Reduce the fraction \frac{20}{6} to lowest terms by extracting and canceling out 2.

$$-\frac{18}{a}+\frac{a}{2}=\frac{10}{3}$$

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,2,3.

$$-6\times 18+3aa=20a$$

Multiply -6 and 18 to get -108.

$$-108+3aa=20a$$

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

$$-108+3a^{2}=20a$$

Subtract 20a from both sides.

$$-108+3a^{2}-20a=0$$

Add 108 to both sides. Anything plus zero gives itself.

$$3a^{2}-20a=108$$

Divide both sides by 3.

$$\frac{3a^{2}-20a}{3}=\frac{108}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$a^{2}-\frac{20}{3}a=\frac{108}{3}$$

Divide 108 by 3.

$$a^{2}-\frac{20}{3}a=36$$

Divide -\frac{20}{3}, the coefficient of the x term, by 2 to get -\frac{10}{3}. Then add the square of -\frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{20}{3}a+\left(-\frac{10}{3}\right)^{2}=36+\left(-\frac{10}{3}\right)^{2}$$

Square -\frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{20}{3}a+\frac{100}{9}=36+\frac{100}{9}$$

Add 36 to \frac{100}{9}.

$$a^{2}-\frac{20}{3}a+\frac{100}{9}=\frac{424}{9}$$

Factor a^{2}-\frac{20}{3}a+\frac{100}{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}.

$$\left(a-\frac{10}{3}\right)^{2}=\frac{424}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{10}{3}\right)^{2}}=\sqrt{\frac{424}{9}}$$

Simplify.

$$a-\frac{10}{3}=\frac{2\sqrt{106}}{3} a-\frac{10}{3}=-\frac{2\sqrt{106}}{3}$$

Add \frac{10}{3} to both sides of the equation.

$$a=\frac{2\sqrt{106}+10}{3} a=\frac{10-2\sqrt{106}}{3}$$