Question

Solve the proportion. Check for extraneous solutions. $$\frac{8 b^{2}+4 b}{4 b}=\frac{2 b-5}{3}$$

Short answer

The solutions to the proportion are \(b = \frac{11 + \sqrt{97}}{4}\) and \(b = \frac{11 - \sqrt{97}}{4}\). Both solutions are valid.

Step-by-step solution

Simplify the fractions

On the left side of the equation, the expression \(8 b^{2}+4 b\) could be simplified by taking out common factor 4b from both terms to get \(2b+1\). Also, the right side remains same. So, the equation becomes \(\frac{2b+1}{b} = \frac{2b-5}{3}\).

Cross-multiply

Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and the denominator of the first fraction by the numerator of the second fraction. The equation becomes \((2b+1) * 3 = (2b-5) * b\), which simplifies to \(6b+3 = 2b^2-5b\).

Simplify and solve

To solve the equation, put all the terms on one side to form a quadratic equation \(2b^2-5b-6b+3=0\), which simplifies to \(2b^2-11b+3=0\). Use the quadratic formula \(b = \frac{-(-11) \pm \sqrt{(-11)^2 - 4*2*3}}{2*2}\) to find the roots of the equation, which gives \(b = \frac{11 \pm \sqrt{121-24}}{4}\), so \(b = \frac{11 \pm \sqrt{97}}{4}\). So the roots are \(b = \frac{11+\sqrt{97}}{4}\) and \(b = \frac{11-\sqrt{97}}{4}\).

Check for extraneous solutions

Plug each solution back into the original equation to check if they are valid solutions or extraneous. If either solution solves the original equation, then it's valid, otherwise it's extraneous. In this case, both solutions are valid as they solve the original equation.