Solve (0.38)^2+0.38B+1/4B^2=0.38

$$( 0.38 ) ^ { 2 } + 0.38 B + \frac { 1 } { 4 } B ^ { 2 } = 0.38$$

Solve for B

$$B=\frac{\sqrt{38}}{5}-0.76\approx 0.472882801$$

$$B=-\frac{\sqrt{38}}{5}-0.76\approx -1.992882801$$

Steps Using the Quadratic Formula

Calculate 0.38 to the power of 2 and get 0.1444.

$$0.1444+0.38B+\frac{1}{4}B^{2}=0.38$$

Subtract 0.38 from both sides.

$$0.1444+0.38B+\frac{1}{4}B^{2}-0.38=0$$

Subtract 0.38 from 0.1444 to get -0.2356.

$$-0.2356+0.38B+\frac{1}{4}B^{2}=0$$

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.

$$\frac{1}{4}B^{2}+0.38B-0.2356=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, 0.38 for b, and -0.2356 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$B=\frac{-0.38±\sqrt{0.38^{2}-4\times \frac{1}{4}\left(-0.2356\right)}}{2\times \frac{1}{4}}$$

Square 0.38 by squaring both the numerator and the denominator of the fraction.

$$B=\frac{-0.38±\sqrt{0.1444-4\times \frac{1}{4}\left(-0.2356\right)}}{2\times \frac{1}{4}}$$

Multiply -4 times \frac{1}{4}.

$$B=\frac{-0.38±\sqrt{0.1444-\left(-0.2356\right)}}{2\times \frac{1}{4}}$$

Multiply -1 times -0.2356.

$$B=\frac{-0.38±\sqrt{\frac{361+589}{2500}}}{2\times \frac{1}{4}}$$

Add 0.1444 to 0.2356 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$B=\frac{-0.38±\sqrt{0.38}}{2\times \frac{1}{4}}$$

Take the square root of 0.38.

$$B=\frac{-0.38±\frac{\sqrt{38}}{10}}{2\times \frac{1}{4}}$$

Multiply 2 times \frac{1}{4}.

$$B=\frac{-0.38±\frac{\sqrt{38}}{10}}{\frac{1}{2}}$$

Now solve the equation B=\frac{-0.38±\frac{\sqrt{38}}{10}}{\frac{1}{2}} when ± is plus. Add -0.38 to \frac{\sqrt{38}}{10}.

$$B=\frac{\frac{\sqrt{38}}{10}-\frac{19}{50}}{\frac{1}{2}}$$

Divide -\frac{19}{50}+\frac{\sqrt{38}}{10} by \frac{1}{2} by multiplying -\frac{19}{50}+\frac{\sqrt{38}}{10} by the reciprocal of \frac{1}{2}.

$$B=\frac{\sqrt{38}}{5}-\frac{19}{25}$$

Now solve the equation B=\frac{-0.38±\frac{\sqrt{38}}{10}}{\frac{1}{2}} when ± is minus. Subtract \frac{\sqrt{38}}{10} from -0.38.

$$B=\frac{-\frac{\sqrt{38}}{10}-\frac{19}{50}}{\frac{1}{2}}$$

Divide -\frac{19}{50}-\frac{\sqrt{38}}{10} by \frac{1}{2} by multiplying -\frac{19}{50}-\frac{\sqrt{38}}{10} by the reciprocal of \frac{1}{2}.

$$B=-\frac{\sqrt{38}}{5}-\frac{19}{25}$$

The equation is now solved.

$$B=\frac{\sqrt{38}}{5}-\frac{19}{25} B=-\frac{\sqrt{38}}{5}-\frac{19}{25}$$

Steps for Completing the Square

Calculate 0.38 to the power of 2 and get 0.1444.

$$0.1444+0.38B+\frac{1}{4}B^{2}=0.38$$

Subtract 0.1444 from both sides.

$$0.38B+\frac{1}{4}B^{2}=0.38-0.1444$$

Subtract 0.1444 from 0.38 to get 0.2356.

$$0.38B+\frac{1}{4}B^{2}=0.2356$$

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.

$$\frac{1}{4}B^{2}+0.38B=0.2356$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}B^{2}+0.38B}{\frac{1}{4}}=\frac{0.2356}{\frac{1}{4}}$$

Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.

$$B^{2}+\frac{0.38}{\frac{1}{4}}B=\frac{0.2356}{\frac{1}{4}}$$

Divide 0.38 by \frac{1}{4} by multiplying 0.38 by the reciprocal of \frac{1}{4}.

$$B^{2}+\frac{38}{25}B=\frac{0.2356}{\frac{1}{4}}$$

Divide 0.2356 by \frac{1}{4} by multiplying 0.2356 by the reciprocal of \frac{1}{4}.

$$B^{2}+\frac{38}{25}B=\frac{589}{625}$$

Divide \frac{38}{25}, the coefficient of the x term, by 2 to get \frac{19}{25}. Then add the square of \frac{19}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$B^{2}+\frac{38}{25}B+\left(\frac{19}{25}\right)^{2}=\frac{589}{625}+\left(\frac{19}{25}\right)^{2}$$

Square \frac{19}{25} by squaring both the numerator and the denominator of the fraction.

$$B^{2}+\frac{38}{25}B+\frac{361}{625}=\frac{589+361}{625}$$

Add \frac{589}{625} to \frac{361}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$B^{2}+\frac{38}{25}B+\frac{361}{625}=\frac{38}{25}$$

Factor B^{2}+\frac{38}{25}B+\frac{361}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(B+\frac{19}{25}\right)^{2}=\frac{38}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(B+\frac{19}{25}\right)^{2}}=\sqrt{\frac{38}{25}}$$

Simplify.

$$B+\frac{19}{25}=\frac{\sqrt{38}}{5} B+\frac{19}{25}=-\frac{\sqrt{38}}{5}$$

Subtract \frac{19}{25} from both sides of the equation.

$$B=\frac{\sqrt{38}}{5}-\frac{19}{25} B=-\frac{\sqrt{38}}{5}-\frac{19}{25}$$