Question

finding x-intercepts in unfactored form
this is the only question in this section.
question
factor to find all x-intercepts of the function.
$f(x) = 2x^2 + x - 6$
answer

Explanation:

Step1: Factor the quadratic function

To factor \( f(x) = 2x^2 + x - 6 \), we look for two numbers that multiply to \( 2\times(-6)= -12 \) and add up to \( 1 \). The numbers are \( 4 \) and \( -3 \) since \( 4\times(-3)= -12 \) and \( 4 + (-3)=1 \).
Rewrite the middle term using these numbers:
\( 2x^2 + 4x - 3x - 6 \)
Group the terms:
\( (2x^2 + 4x) + (-3x - 6) \)
Factor out the common factors from each group:
\( 2x(x + 2) - 3(x + 2) \)
Now, factor out \( (x + 2) \):
\( (2x - 3)(x + 2) \)

Step2: Find the x-intercepts

To find the x-intercepts, set \( f(x) = 0 \):
\( (2x - 3)(x + 2) = 0 \)
Use the zero - product property, which states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).
Case 1: \( 2x - 3 = 0 \)
Solve for \( x \):
\( 2x=3 \)
\( x=\frac{3}{2} \)
Case 2: \( x + 2 = 0 \)
Solve for \( x \):
\( x=-2 \)

Answer:

The x - intercepts are \( x = - 2 \) and \( x=\frac{3}{2} \)