At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

What are the [tex]$x$[/tex]-intercepts of the graph of the function [tex]$f(x) = x^2 + 4x - 12$[/tex]?

A. [tex]$(-6, 0), (2, 0)$[/tex]
B. [tex]$(-2, -16), (0, -12)$[/tex]
C. [tex]$(-6, 0), (-2, -16), (2, 0)$[/tex]
D. [tex]$(0, -12), (-6, 0), (2, 0)$[/tex]


Sagot :

To find the [tex]$x$[/tex]-intercepts of the graph of the function [tex]\( f(x) = x^2 + 4x - 12 \)[/tex], we need to determine the points where the graph intersects the x-axis. At the x-intercepts, the value of [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) is zero. Therefore, we need to solve the equation:

[tex]\[ x^2 + 4x - 12 = 0 \][/tex]

This is a quadratic equation, so we can solve for [tex]\( x \)[/tex] using the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -12 \)[/tex]

Plugging the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:

[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \][/tex]

Calculate the discriminant:

[tex]\[ 4^2 - 4 \cdot 1 \cdot (-12) = 16 + 48 = 64 \][/tex]

So, the formula now becomes:

[tex]\[ x = \frac{-4 \pm \sqrt{64}}{2} \][/tex]

Simplify further:

[tex]\[ x = \frac{-4 \pm 8}{2} \][/tex]

This gives us two solutions:

[tex]\[ x = \frac{-4 + 8}{2} = \frac{4}{2} = 2 \][/tex]

and

[tex]\[ x = \frac{-4 - 8}{2} = \frac{-12}{2} = -6 \][/tex]

Thus, the solutions to the equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = -6 \)[/tex]. These solutions represent the x-intercepts of the graph of the function [tex]\( f(x) \)[/tex].

Therefore, the x-intercepts are at the points [tex]\( (-6, 0) \)[/tex] and [tex]\( (2, 0) \)[/tex].

So, the correct answer is:
[tex]\[ (-6, 0),(2, 0) \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.