Solve {l}{2b+15}{-3b^2}

$$\left. \begin{array} { l } { 2 b + 15 } \\ { - 3 b ^ { 2 } } \end{array} \right.$$

Factor

$$-3\left(b-\frac{1-\sqrt{46}}{3}\right)\left(b-\frac{\sqrt{46}+1}{3}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-3b^{2}+2b+15=0$$

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.

$$b=\frac{-2±\sqrt{2^{2}-4\left(-3\right)\times 15}}{2\left(-3\right)}$$

Square 2.

$$b=\frac{-2±\sqrt{4-4\left(-3\right)\times 15}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$b=\frac{-2±\sqrt{4+12\times 15}}{2\left(-3\right)}$$

Multiply 12 times 15.

$$b=\frac{-2±\sqrt{4+180}}{2\left(-3\right)}$$

Add 4 to 180.

$$b=\frac{-2±\sqrt{184}}{2\left(-3\right)}$$

Take the square root of 184.

$$b=\frac{-2±2\sqrt{46}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$b=\frac{-2±2\sqrt{46}}{-6}$$

Now solve the equation b=\frac{-2±2\sqrt{46}}{-6} when ± is plus. Add -2 to 2\sqrt{46}.

$$b=\frac{2\sqrt{46}-2}{-6}$$

Divide -2+2\sqrt{46} by -6.

$$b=\frac{1-\sqrt{46}}{3}$$

Now solve the equation b=\frac{-2±2\sqrt{46}}{-6} when ± is minus. Subtract 2\sqrt{46} from -2.

$$b=\frac{-2\sqrt{46}-2}{-6}$$

Divide -2-2\sqrt{46} by -6.

$$b=\frac{\sqrt{46}+1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{46}}{3} for x_{1} and \frac{1+\sqrt{46}}{3} for x_{2}.

$$-3b^{2}+2b+15=-3\left(b-\frac{1-\sqrt{46}}{3}\right)\left(b-\frac{\sqrt{46}+1}{3}\right)$$

Evaluate

$$15+2b-3b^{2}$$