To find the [tex]$x$[/tex]-intercepts of the function [tex]\(f(x) = 15x^2 - 12x - 48\)[/tex], we need to find the values of [tex]\(x\)[/tex] where the function equals zero, i.e., solve the equation [tex]\(15x^2 - 12x - 48 = 0\)[/tex].
Step 1: Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 15\)[/tex], [tex]\(b = -12\)[/tex], and [tex]\(c = -48\)[/tex].
Step 2: Calculate the discriminant [tex]\(\Delta\)[/tex], given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula.
Step 3: Use the quadratic formula to find the roots of the equation. The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant [tex]\(\Delta\)[/tex] into the formula to find the two solutions for [tex]\(x\)[/tex].
Thus, the solutions to the quadratic equation are:
[tex]\[
x_1 = \frac{-(-12) + \sqrt{3240}}{2 \cdot 15} = \frac{12 + \sqrt{3240}}{30}
\][/tex]
[tex]\[
x_2 = \frac{-(-12) - \sqrt{3240}}{2 \cdot 15} = \frac{12 - \sqrt{3240}}{30}
\][/tex]
Step 4: Simplify the solutions:
[tex]\[
x_1 \approx 2.23
\][/tex]
[tex]\[
x_2 \approx -1.43
\][/tex]
Therefore, the [tex]$x$[/tex]-intercepts of the function [tex]\(f(x) = 15x^2 - 12x - 48\)[/tex] are approximately [tex]\((2.23, 0)\)[/tex] and [tex]\((-1.43, 0)\)[/tex].
The correct answer choice is:
b) [tex]\((2.23, 0), (-1.43, 0)\)[/tex]
