Solve 10b^2=27b-181

$$10 b ^ { 2 } = 27 b - 181$$

Solve for b

$$b=\frac{27+\sqrt{6511}i}{20}\approx 1.35+4.034538387i$$

$$b=\frac{-\sqrt{6511}i+27}{20}\approx 1.35-4.034538387i$$

Steps Using the Quadratic Formula

Subtract 27b from both sides.

$$10b^{2}-27b=-181$$

Add 181 to both sides.

$$10b^{2}-27b+181=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -27 for b, and 181 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 10\times 181}}{2\times 10}$$

Square -27.

$$b=\frac{-\left(-27\right)±\sqrt{729-4\times 10\times 181}}{2\times 10}$$

Multiply -4 times 10.

$$b=\frac{-\left(-27\right)±\sqrt{729-40\times 181}}{2\times 10}$$

Multiply -40 times 181.

$$b=\frac{-\left(-27\right)±\sqrt{729-7240}}{2\times 10}$$

Add 729 to -7240.

$$b=\frac{-\left(-27\right)±\sqrt{-6511}}{2\times 10}$$

Take the square root of -6511.

$$b=\frac{-\left(-27\right)±\sqrt{6511}i}{2\times 10}$$

The opposite of -27 is 27.

$$b=\frac{27±\sqrt{6511}i}{2\times 10}$$

Multiply 2 times 10.

$$b=\frac{27±\sqrt{6511}i}{20}$$

Now solve the equation b=\frac{27±\sqrt{6511}i}{20} when ± is plus. Add 27 to i\sqrt{6511}.

$$b=\frac{27+\sqrt{6511}i}{20}$$

Now solve the equation b=\frac{27±\sqrt{6511}i}{20} when ± is minus. Subtract i\sqrt{6511} from 27.

$$b=\frac{-\sqrt{6511}i+27}{20}$$

The equation is now solved.

$$b=\frac{27+\sqrt{6511}i}{20} b=\frac{-\sqrt{6511}i+27}{20}$$

Steps for Completing the Square

Subtract 27b from both sides.

$$10b^{2}-27b=-181$$

Divide both sides by 10.

$$\frac{10b^{2}-27b}{10}=-\frac{181}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$b^{2}-\frac{27}{10}b=-\frac{181}{10}$$

Divide -\frac{27}{10}, the coefficient of the x term, by 2 to get -\frac{27}{20}. Then add the square of -\frac{27}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-\frac{27}{10}b+\left(-\frac{27}{20}\right)^{2}=-\frac{181}{10}+\left(-\frac{27}{20}\right)^{2}$$

Square -\frac{27}{20} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-\frac{27}{10}b+\frac{729}{400}=-\frac{181}{10}+\frac{729}{400}$$

Add -\frac{181}{10} to \frac{729}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$b^{2}-\frac{27}{10}b+\frac{729}{400}=-\frac{6511}{400}$$

Factor b^{2}-\frac{27}{10}b+\frac{729}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{27}{20}\right)^{2}=-\frac{6511}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{27}{20}\right)^{2}}=\sqrt{-\frac{6511}{400}}$$

Simplify.

$$b-\frac{27}{20}=\frac{\sqrt{6511}i}{20} b-\frac{27}{20}=-\frac{\sqrt{6511}i}{20}$$

Add \frac{27}{20} to both sides of the equation.

$$b=\frac{27+\sqrt{6511}i}{20} b=\frac{-\sqrt{6511}i+27}{20}$$