Question

factor $k^2 - 10k + 24$. the factored expression is

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(24\) (the constant term) and add up to \(-10\) (the coefficient of the middle term). Let's list the factor pairs of \(24\): \((1,24)\), \((2,12)\), \((3,8)\), \((4,6)\). Now, check which pair adds up to \(10\) (we consider negative numbers since the sum is \(-10\) and product is positive \(24\), so both numbers are negative). The pair \( -4\) and \( -6\) multiply to \(24\) (\((-4)\times(-6) = 24\)) and add up to \(-10\) (\((-4)+(-6)=-10\)).

Step2: Factor the quadratic

Using the numbers we found, we can factor the quadratic \(k^{2}-10k + 24\) as \((k - 4)(k - 6)\). We use the formula for factoring a quadratic \(ax^{2}+bx + c\) (here \(a = 1\)): \(x^{2}+bx + c=(x + m)(x + n)\) where \(m\) and \(n\) are the two numbers we found. Substituting \(m=-4\) and \(n = -6\), we get \((k-4)(k - 6)\).

Answer:

\((k - 4)(k - 6)\)