Question

factor completely.
( k^2 - 9k - 22 )

Explanation:

Step1: Identify the quadratic form

We have the quadratic expression \(k^{2}-9k - 22\). For a quadratic of the form \(ax^{2}+bx + c\) (here \(a = 1\), \(b=-9\), \(c = - 22\)), we need to find two numbers that multiply to \(a\times c=1\times(-22)=-22\) and add up to \(b=-9\).

Step2: Find the two numbers

We need two numbers \(m\) and \(n\) such that \(m\times n=-22\) and \(m + n=-9\). The numbers are \(-11\) and \(2\) since \((-11)\times2=-22\) and \(-11 + 2=-9\).

Step3: Rewrite the middle term

Rewrite the quadratic as \(k^{2}-11k+2k - 22\).

Step4: Factor by grouping

Group the first two terms and the last two terms: \((k^{2}-11k)+(2k - 22)\). Factor out the greatest common factor from each group: \(k(k - 11)+2(k - 11)\). Then factor out the common binomial factor \((k - 11)\): \((k - 11)(k + 2)\).

Answer:

\((k - 11)(k + 2)\)