Solve -6z+1z^2-27

$$- 6 z + 1 z ^ { 2 } - 27$$

Factor

$$\left(z-9\right)\left(z+3\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$z^{2}-6z-27$$

Factor the expression by grouping. First, the expression needs to be rewritten as z^{2}+az+bz-27. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-27\right)=-27$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -27.

$$1,-27 3,-9$$

Calculate the sum for each pair.

$$1-27=-26 3-9=-6$$

The solution is the pair that gives sum -6.

$$a=-9 b=3$$

Rewrite z^{2}-6z-27 as \left(z^{2}-9z\right)+\left(3z-27\right).

$$\left(z^{2}-9z\right)+\left(3z-27\right)$$

Factor out z in the first and 3 in the second group.

$$z\left(z-9\right)+3\left(z-9\right)$$

Factor out common term z-9 by using distributive property.

$$\left(z-9\right)\left(z+3\right)$$

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.

$$z^{2}-6z-27=0$$

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.

$$z=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-27\right)}}{2}$$

Square -6.

$$z=\frac{-\left(-6\right)±\sqrt{36-4\left(-27\right)}}{2}$$

Multiply -4 times -27.

$$z=\frac{-\left(-6\right)±\sqrt{36+108}}{2}$$

Add 36 to 108.

$$z=\frac{-\left(-6\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$z=\frac{-\left(-6\right)±12}{2}$$

The opposite of -6 is 6.

$$z=\frac{6±12}{2}$$

Now solve the equation z=\frac{6±12}{2} when ± is plus. Add 6 to 12.

$$z=\frac{18}{2}$$

Divide 18 by 2.

$$z=9$$

Now solve the equation z=\frac{6±12}{2} when ± is minus. Subtract 12 from 6.

$$z=-\frac{6}{2}$$

Divide -6 by 2.

$$z=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -3 for x_{2}.

$$z^{2}-6z-27=\left(z-9\right)\left(z-\left(-3\right)\right)$$

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

$$z^{2}-6z-27=\left(z-9\right)\left(z+3\right)$$

Evaluate

$$\left(z-9\right)\left(z+3\right)$$