Solve 6a^2+9a

$$6 a ^ { 2 } + 9 a$$

Factor

$$3a\left(2a+3\right)$$

Solution Steps

Factor out 3.

$$3\left(2a^{2}+3a\right)$$

Consider 2a^{2}+3a. Factor out a.

$$a\left(2a+3\right)$$

Rewrite the complete factored expression.

$$3a\left(2a+3\right)$$

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.

$$6a^{2}+9a=0$$

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.

$$a=\frac{-9±\sqrt{9^{2}}}{2\times 6}$$

Take the square root of 9^{2}.

$$a=\frac{-9±9}{2\times 6}$$

Multiply 2 times 6.

$$a=\frac{-9±9}{12}$$

Now solve the equation a=\frac{-9±9}{12} when ± is plus. Add -9 to 9.

$$a=\frac{0}{12}$$

Divide 0 by 12.

$$a=0$$

Now solve the equation a=\frac{-9±9}{12} when ± is minus. Subtract 9 from -9.

$$a=-\frac{18}{12}$$

Reduce the fraction \frac{-18}{12} to lowest terms by extracting and canceling out 6.

$$a=-\frac{3}{2}$$

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{3}{2} for x_{2}.

$$6a^{2}+9a=6a\left(a-\left(-\frac{3}{2}\right)\right)$$

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

$$6a^{2}+9a=6a\left(a+\frac{3}{2}\right)$$

Add \frac{3}{2} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6a^{2}+9a=6a\times \frac{2a+3}{2}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$6a^{2}+9a=3a\left(2a+3\right)$$

Evaluate

$$3a\left(2a+3\right)$$