Solve 8a+8a-172-2168/a=560

$$8 a + 8 a - 172 - 2 \frac { 168 } { a } = 560$$

Solve for a

$$a = \frac{\sqrt{34833} + 183}{8} \approx 46.204501173$$

$$a=\frac{183-\sqrt{34833}}{8}\approx -0.454501173$$

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 a.

$$a\left(8a+8a-172\right)-2\times 168=560a$$

Combine 8a and 8a to get 16a.

$$a\left(16a-172\right)-2\times 168=560a$$

Use the distributive property to multiply a by 16a-172.

$$16a^{2}-172a-2\times 168=560a$$

Multiply 2 and 168 to get 336.

$$16a^{2}-172a-336=560a$$

Subtract 560a from both sides.

$$16a^{2}-172a-336-560a=0$$

Combine -172a and -560a to get -732a.

$$16a^{2}-732a-336=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -732 for b, and -336 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-732\right)±\sqrt{\left(-732\right)^{2}-4\times 16\left(-336\right)}}{2\times 16}$$

Square -732.

$$a=\frac{-\left(-732\right)±\sqrt{535824-4\times 16\left(-336\right)}}{2\times 16}$$

Multiply -4 times 16.

$$a=\frac{-\left(-732\right)±\sqrt{535824-64\left(-336\right)}}{2\times 16}$$

Multiply -64 times -336.

$$a=\frac{-\left(-732\right)±\sqrt{535824+21504}}{2\times 16}$$

Add 535824 to 21504.

$$a=\frac{-\left(-732\right)±\sqrt{557328}}{2\times 16}$$

Take the square root of 557328.

$$a=\frac{-\left(-732\right)±4\sqrt{34833}}{2\times 16}$$

The opposite of -732 is 732.

$$a=\frac{732±4\sqrt{34833}}{2\times 16}$$

Multiply 2 times 16.

$$a=\frac{732±4\sqrt{34833}}{32}$$

Now solve the equation a=\frac{732±4\sqrt{34833}}{32} when ± is plus. Add 732 to 4\sqrt{34833}.

$$a=\frac{4\sqrt{34833}+732}{32}$$

Divide 732+4\sqrt{34833} by 32.

$$a=\frac{\sqrt{34833}+183}{8}$$

Now solve the equation a=\frac{732±4\sqrt{34833}}{32} when ± is minus. Subtract 4\sqrt{34833} from 732.

$$a=\frac{732-4\sqrt{34833}}{32}$$

Divide 732-4\sqrt{34833} by 32.

$$a=\frac{183-\sqrt{34833}}{8}$$

The equation is now solved.

$$a=\frac{\sqrt{34833}+183}{8} a=\frac{183-\sqrt{34833}}{8}$$

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 a.

$$a\left(8a+8a-172\right)-2\times 168=560a$$

Combine 8a and 8a to get 16a.

$$a\left(16a-172\right)-2\times 168=560a$$

Use the distributive property to multiply a by 16a-172.

$$16a^{2}-172a-2\times 168=560a$$

Multiply 2 and 168 to get 336.

$$16a^{2}-172a-336=560a$$

Subtract 560a from both sides.

$$16a^{2}-172a-336-560a=0$$

Combine -172a and -560a to get -732a.

$$16a^{2}-732a-336=0$$

Add 336 to both sides. Anything plus zero gives itself.

$$16a^{2}-732a=336$$

Divide both sides by 16.

$$\frac{16a^{2}-732a}{16}=\frac{336}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$a^{2}+\left(-\frac{732}{16}\right)a=\frac{336}{16}$$

Reduce the fraction \frac{-732}{16} to lowest terms by extracting and canceling out 4.

$$a^{2}-\frac{183}{4}a=\frac{336}{16}$$

Divide 336 by 16.

$$a^{2}-\frac{183}{4}a=21$$

Divide -\frac{183}{4}, the coefficient of the x term, by 2 to get -\frac{183}{8}. Then add the square of -\frac{183}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{183}{4}a+\left(-\frac{183}{8}\right)^{2}=21+\left(-\frac{183}{8}\right)^{2}$$

Square -\frac{183}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{183}{4}a+\frac{33489}{64}=21+\frac{33489}{64}$$

Add 21 to \frac{33489}{64}.

$$a^{2}-\frac{183}{4}a+\frac{33489}{64}=\frac{34833}{64}$$

Factor a^{2}-\frac{183}{4}a+\frac{33489}{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}.

$$\left(a-\frac{183}{8}\right)^{2}=\frac{34833}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{183}{8}\right)^{2}}=\sqrt{\frac{34833}{64}}$$

Simplify.

$$a-\frac{183}{8}=\frac{\sqrt{34833}}{8} a-\frac{183}{8}=-\frac{\sqrt{34833}}{8}$$

Add \frac{183}{8} to both sides of the equation.

$$a=\frac{\sqrt{34833}+183}{8} a=\frac{183-\sqrt{34833}}{8}$$