Question

which expression is equivalent to the given expression?
$2x^2 - 14x + 24$

a. $2(x - 5)(x - 2)$

b. $(2x - 12)(x - 2)$

c. $2(x - 8)(x + 3)$

d. $2(x - 3)(x - 4)$

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from the quadratic expression \(2x^2 - 14x + 24\). The GCF of 2, -14, and 24 is 2. So we get:
\(2(x^2 - 7x + 12)\)

Step2: Factor the quadratic inside the parentheses

Now, we need to factor the quadratic \(x^2 - 7x + 12\). We look for two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of the x-term). The numbers -3 and -4 satisfy this because \((-3) \times (-4) = 12\) and \((-3) + (-4) = -7\). So, \(x^2 - 7x + 12\) factors as \((x - 3)(x - 4)\).

Step3: Combine the factors

Putting it all together, the original expression \(2x^2 - 14x + 24\) factors to \(2(x - 3)(x - 4)\).

Answer:

D. \(2(x - 3)(x - 4)\)