Solve 3z^2+5z)+(z^2+2z)+(z-4)

$$3 z ^ { 2 } + 5 z ) + ( z ^ { 2 } + 2 z ) + ( z - 4 )$$

Factor

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

Solution Steps

Factor out z.

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

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.

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

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.

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

Take the square root of 5^{2}.

$$z=\frac{-5±5}{2\times 3}$$

Multiply 2 times 3.

$$z=\frac{-5±5}{6}$$

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

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

Divide 0 by 6.

$$z=0$$

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

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

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

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

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}.

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

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

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

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

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

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

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

Evaluate

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