Question

factor each expression:
$x^2 - 13x + 42 = $
$x^2 - 2x - 24 = $
$x^2 + 5x - 14 = $
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Explanation:

Step1: Factor \(x^2 - 13x + 42\)

We need two numbers that multiply to \(42\) and add up to \(-13\). The numbers are \(-6\) and \(-7\) since \((-6)\times(-7)=42\) and \((-6)+(-7)=-13\). So, \(x^2 - 13x + 42=(x - 6)(x - 7)\).

Step2: Factor \(x^2 - 2x - 24\)

We need two numbers that multiply to \(-24\) and add up to \(-2\). The numbers are \(-6\) and \(4\) since \((-6)\times4=-24\) and \((-6)+4=-2\). So, \(x^2 - 2x - 24=(x - 6)(x + 4)\).

Step3: Factor \(x^2 + 5x - 14\)

We need two numbers that multiply to \(-14\) and add up to \(5\). The numbers are \(7\) and \(-2\) since \(7\times(-2)=-14\) and \(7+(-2)=5\). So, \(x^2 + 5x - 14=(x + 7)(x - 2)\).

Answer:

\(x^2 - 13x + 42=(x - 6)(x - 7)\)
\(x^2 - 2x - 24=(x - 6)(x + 4)\)
\(x^2 + 5x - 14=(x + 7)(x - 2)\)