Question

cd: factoring a=1
possible points: 2
show how to rewrite $x^2 - 11x + 30$ in factored form.
factors: ( ) and ( )

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(30\) (the constant term) and add up to \(-11\) (the coefficient of the middle term). Let's list the factor pairs of \(30\): \(1\) and \(30\), \(2\) and \(15\), \(3\) and \(10\), \(5\) and \(6\). Since the middle term is \(-11x\) and the constant term is \(+30\), both numbers should be negative. The pair \(-5\) and \(-6\) multiply to \(30\) (\((-5)\times(-6) = 30\)) and add up to \(-11\) (\(-5 + (-6)=-11\)).

Step2: Factor the quadratic

Using the numbers we found, we can factor \(x^{2}-11x + 30\) as \((x - 5)(x - 6)\). We check this by expanding: \((x - 5)(x - 6)=x^{2}-6x-5x + 30=x^{2}-11x + 30\), which matches the original quadratic.

Answer:

\((x - 5)\) and \((x - 6)\)