Solve 9z+30z^2

$$9 z + 30 z ^ { 2 }$$

Factor

$$3z\left(10z+3\right)$$

Solution Steps

Factor out 3.

$$3\left(3z+10z^{2}\right)$$

Consider 3z+10z^{2}. Factor out z.

$$z\left(3+10z\right)$$

Rewrite the complete factored expression.

$$3z\left(10z+3\right)$$

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.

$$30z^{2}+9z=0$$

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.

$$z=\frac{-9±\sqrt{9^{2}}}{2\times 30}$$

Take the square root of 9^{2}.

$$z=\frac{-9±9}{2\times 30}$$

Multiply 2 times 30.

$$z=\frac{-9±9}{60}$$

Now solve the equation z=\frac{-9±9}{60} when ± is plus. Add -9 to 9.

$$z=\frac{0}{60}$$

Divide 0 by 60.

$$z=0$$

Now solve the equation z=\frac{-9±9}{60} when ± is minus. Subtract 9 from -9.

$$z=-\frac{18}{60}$$

Reduce the fraction \frac{-18}{60} to lowest terms by extracting and canceling out 6.

$$z=-\frac{3}{10}$$

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}{10} for x_{2}.

$$30z^{2}+9z=30z\left(z-\left(-\frac{3}{10}\right)\right)$$

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

$$30z^{2}+9z=30z\left(z+\frac{3}{10}\right)$$

Add \frac{3}{10} to z by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$30z^{2}+9z=30z\times \frac{10z+3}{10}$$

Cancel out 10, the greatest common factor in 30 and 10.

$$30z^{2}+9z=3z\left(10z+3\right)$$

Evaluate

$$3z\left(10z+3\right)$$