Solve q^2-5q-24

$${ q }^{ 2 } -5q-24$$

Factor

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

Solution Steps

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

$$a+b=-5 ab=1\left(-24\right)=-24$$

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

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

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

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

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

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

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.

$$q^{2}-5q-24=0$$

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.

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

Square -5.

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

Multiply -4 times -24.

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

Add 25 to 96.

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

Take the square root of 121.

$$q=\frac{-\left(-5\right)±11}{2}$$

The opposite of -5 is 5.

$$q=\frac{5±11}{2}$$

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

$$q=\frac{16}{2}$$

Divide 16 by 2.

$$q=8$$

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

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

Divide -6 by 2.

$$q=-3$$

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

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

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

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

Evaluate

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