Solve 80b/13=5b^2+1000/169-5

$$\frac { 80 b } { 13 } = 5 b ^ { 2 } + \frac { 1000 } { 169 } - 5$$

Solve for b

$$b = \frac{\sqrt{33} + 8}{13} \approx 1.05727405$$

$$b=\frac{8-\sqrt{33}}{13}\approx 0.173495181$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 169, the least common multiple of 13,169.

$$13\times 80b=845b^{2}+1000-845$$

Multiply 13 and 80 to get 1040.

$$1040b=845b^{2}+1000-845$$

Subtract 845 from 1000 to get 155.

$$1040b=845b^{2}+155$$

Subtract 845b^{2} from both sides.

$$1040b-845b^{2}=155$$

Subtract 155 from both sides.

$$1040b-845b^{2}-155=0$$

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.

$$-845b^{2}+1040b-155=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -845 for a, 1040 for b, and -155 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-1040±\sqrt{1040^{2}-4\left(-845\right)\left(-155\right)}}{2\left(-845\right)}$$

Square 1040.

$$b=\frac{-1040±\sqrt{1081600-4\left(-845\right)\left(-155\right)}}{2\left(-845\right)}$$

Multiply -4 times -845.

$$b=\frac{-1040±\sqrt{1081600+3380\left(-155\right)}}{2\left(-845\right)}$$

Multiply 3380 times -155.

$$b=\frac{-1040±\sqrt{1081600-523900}}{2\left(-845\right)}$$

Add 1081600 to -523900.

$$b=\frac{-1040±\sqrt{557700}}{2\left(-845\right)}$$

Take the square root of 557700.

$$b=\frac{-1040±130\sqrt{33}}{2\left(-845\right)}$$

Multiply 2 times -845.

$$b=\frac{-1040±130\sqrt{33}}{-1690}$$

Now solve the equation b=\frac{-1040±130\sqrt{33}}{-1690} when ± is plus. Add -1040 to 130\sqrt{33}.

$$b=\frac{130\sqrt{33}-1040}{-1690}$$

Divide -1040+130\sqrt{33} by -1690.

$$b=\frac{8-\sqrt{33}}{13}$$

Now solve the equation b=\frac{-1040±130\sqrt{33}}{-1690} when ± is minus. Subtract 130\sqrt{33} from -1040.

$$b=\frac{-130\sqrt{33}-1040}{-1690}$$

Divide -1040-130\sqrt{33} by -1690.

$$b=\frac{\sqrt{33}+8}{13}$$

The equation is now solved.

$$b=\frac{8-\sqrt{33}}{13} b=\frac{\sqrt{33}+8}{13}$$

Steps for Completing the Square

Multiply both sides of the equation by 169, the least common multiple of 13,169.

$$13\times 80b=845b^{2}+1000-845$$

Multiply 13 and 80 to get 1040.

$$1040b=845b^{2}+1000-845$$

Subtract 845 from 1000 to get 155.

$$1040b=845b^{2}+155$$

Subtract 845b^{2} from both sides.

$$1040b-845b^{2}=155$$

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.

$$-845b^{2}+1040b=155$$

Divide both sides by -845.

$$\frac{-845b^{2}+1040b}{-845}=\frac{155}{-845}$$

Dividing by -845 undoes the multiplication by -845.

$$b^{2}+\frac{1040}{-845}b=\frac{155}{-845}$$

Reduce the fraction \frac{1040}{-845} to lowest terms by extracting and canceling out 65.

$$b^{2}-\frac{16}{13}b=\frac{155}{-845}$$

Reduce the fraction \frac{155}{-845} to lowest terms by extracting and canceling out 5.

$$b^{2}-\frac{16}{13}b=-\frac{31}{169}$$

Divide -\frac{16}{13}, the coefficient of the x term, by 2 to get -\frac{8}{13}. Then add the square of -\frac{8}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-\frac{16}{13}b+\left(-\frac{8}{13}\right)^{2}=-\frac{31}{169}+\left(-\frac{8}{13}\right)^{2}$$

Square -\frac{8}{13} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-\frac{16}{13}b+\frac{64}{169}=\frac{-31+64}{169}$$

Add -\frac{31}{169} to \frac{64}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$b^{2}-\frac{16}{13}b+\frac{64}{169}=\frac{33}{169}$$

Factor b^{2}-\frac{16}{13}b+\frac{64}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{8}{13}\right)^{2}=\frac{33}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{8}{13}\right)^{2}}=\sqrt{\frac{33}{169}}$$

Simplify.

$$b-\frac{8}{13}=\frac{\sqrt{33}}{13} b-\frac{8}{13}=-\frac{\sqrt{33}}{13}$$

Add \frac{8}{13} to both sides of the equation.

$$b=\frac{\sqrt{33}+8}{13} b=\frac{8-\sqrt{33}}{13}$$