Solve 24+5z-z^2

$$24 + 5 z - z ^ { 2 }$$

Factor

$$-\left(z-8\right)\left(z+3\right)$$

Solution Steps

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

$$-z^{2}+5z+24$$

Factor the expression by grouping. First, the expression needs to be rewritten as -z^{2}+az+bz+24. To find a and b, set up a system to be solved.

$$a+b=5 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=8 b=-3$$

Rewrite -z^{2}+5z+24 as \left(-z^{2}+8z\right)+\left(-3z+24\right).

$$\left(-z^{2}+8z\right)+\left(-3z+24\right)$$

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

$$-z\left(z-8\right)-3\left(z-8\right)$$

Factor out common term z-8 by using distributive property.

$$\left(z-8\right)\left(-z-3\right)$$

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.

$$-z^{2}+5z+24=0$$

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.

$$z=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 5.

$$z=\frac{-5±\sqrt{25-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$z=\frac{-5±\sqrt{25+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$z=\frac{-5±\sqrt{25+96}}{2\left(-1\right)}$$

Add 25 to 96.

$$z=\frac{-5±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$z=\frac{-5±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{-5±11}{-2}$$

Now solve the equation z=\frac{-5±11}{-2} when ± is plus. Add -5 to 11.

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

Divide 6 by -2.

$$z=-3$$

Now solve the equation z=\frac{-5±11}{-2} when ± is minus. Subtract 11 from -5.

$$z=-\frac{16}{-2}$$

Divide -16 by -2.

$$z=8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and 8 for x_{2}.

$$-z^{2}+5z+24=-\left(z-\left(-3\right)\right)\left(z-8\right)$$

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

$$-z^{2}+5z+24=-\left(z+3\right)\left(z-8\right)$$

Evaluate

$$-\left(z-8\right)\left(z+3\right)$$