Solve 12b^2+15b

$$12 b ^ { 2 } + 15 b$$

Factor

$$3b\left(4b+5\right)$$

Solution Steps

Factor out 3.

$$3\left(4b^{2}+5b\right)$$

Consider 4b^{2}+5b. Factor out b.

$$b\left(4b+5\right)$$

Rewrite the complete factored expression.

$$3b\left(4b+5\right)$$

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.

$$12b^{2}+15b=0$$

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.

$$b=\frac{-15±\sqrt{15^{2}}}{2\times 12}$$

Take the square root of 15^{2}.

$$b=\frac{-15±15}{2\times 12}$$

Multiply 2 times 12.

$$b=\frac{-15±15}{24}$$

Now solve the equation b=\frac{-15±15}{24} when ± is plus. Add -15 to 15.

$$b=\frac{0}{24}$$

Divide 0 by 24.

$$b=0$$

Now solve the equation b=\frac{-15±15}{24} when ± is minus. Subtract 15 from -15.

$$b=-\frac{30}{24}$$

Reduce the fraction \frac{-30}{24} to lowest terms by extracting and canceling out 6.

$$b=-\frac{5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -\frac{5}{4} for x_{2}.

$$12b^{2}+15b=12b\left(b-\left(-\frac{5}{4}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$12b^{2}+15b=12b\left(b+\frac{5}{4}\right)$$

Add \frac{5}{4} to b by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$12b^{2}+15b=12b\times \frac{4b+5}{4}$$

Cancel out 4, the greatest common factor in 12 and 4.

$$12b^{2}+15b=3b\left(4b+5\right)$$

Evaluate

$$3b\left(4b+5\right)$$