What are the [tex]$x$[/tex]-intercepts of the function [tex]$f(x) = 15x^2 - 12x - 48$[/tex]?

A. [tex]$(4.47, 0), (-2.87, 0)$[/tex]

B. [tex]$(2.23, 0), (-1.43, 0)$[/tex]

C. [tex]$(33.5, 0), (-21.5, 0)$[/tex]

D. [tex]$(0.4, 0), (-50.4, 0)$[/tex]



Answer :

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]