Solve P(M)=3M^2+5M

$$P ( M ) = 3 M ^ { 2 } + 5 M$$

Factor

$$M\left(3M+5\right)$$

Solution Steps

Factor out M.

$$M\left(3M+5\right)$$

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.

$$3M^{2}+5M=0$$

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.

$$M=\frac{-5±\sqrt{5^{2}}}{2\times 3}$$

Take the square root of 5^{2}.

$$M=\frac{-5±5}{2\times 3}$$

Multiply 2 times 3.

$$M=\frac{-5±5}{6}$$

Now solve the equation M=\frac{-5±5}{6} when ± is plus. Add -5 to 5.

$$M=\frac{0}{6}$$

Divide 0 by 6.

$$M=0$$

Now solve the equation M=\frac{-5±5}{6} when ± is minus. Subtract 5 from -5.

$$M=-\frac{10}{6}$$

Reduce the fraction \frac{-10}{6} to lowest terms by extracting and canceling out 2.

$$M=-\frac{5}{3}$$

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}{3} for x_{2}.

$$3M^{2}+5M=3M\left(M-\left(-\frac{5}{3}\right)\right)$$

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

$$3M^{2}+5M=3M\left(M+\frac{5}{3}\right)$$

Add \frac{5}{3} to M by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3M^{2}+5M=3M\times \frac{3M+5}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3M^{2}+5M=M\left(3M+5\right)$$

Evaluate

$$M\left(3M+5\right)$$