Question

question
factor to find all x-intercepts of the function.
$f(x) = 2x^2 - 4x - 30$
answer attempt 1 out of 2
additional solution skip solution

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from the quadratic expression. The GCF of \(2x^2\), \(-4x\), and \(-30\) is 2. So we have:
\(f(x) = 2(x^2 - 2x - 15)\)

Step2: Factor the quadratic trinomial

Now, factor the quadratic trinomial \(x^2 - 2x - 15\). We need two numbers that multiply to \(-15\) and add up to \(-2\). The numbers are \(-5\) and \(3\) because \(-5 \times 3 = -15\) and \(-5 + 3 = -2\). So we can factor it as:
\(x^2 - 2x - 15 = (x - 5)(x + 3)\)

Step3: Write the fully factored form

Substitute the factored trinomial back into the expression from Step 1:
\(f(x) = 2(x - 5)(x + 3)\)

Step4: Find the x-intercepts

To find the x-intercepts, set \(f(x) = 0\):
\(2(x - 5)(x + 3) = 0\)
Since 2 is never zero, we set each factor equal to zero:
\(x - 5 = 0\) or \(x + 3 = 0\)
Solving these equations gives \(x = 5\) or \(x = -3\)

Answer:

The x-intercepts of the function \(f(x) = 2x^2 - 4x - 30\) are \(x = -3\) and \(x = 5\).