To find the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 - 6x + 8 \)[/tex], we start by setting the function equal to zero:
[tex]\[ x^2 - 6x + 8 = 0 \][/tex]
We need to find the values of [tex]\( x \)[/tex] that satisfy this equation. For a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the solutions can be determined using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this equation, we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -6 \)[/tex]
- [tex]\( c = 8 \)[/tex]
First, calculate the discriminant ([tex]\( \Delta \)[/tex]) which is under the square root in the quadratic formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-6)^2 - 4 \cdot 1 \cdot 8 \][/tex]
[tex]\[ \Delta = 36 - 32 \][/tex]
[tex]\[ \Delta = 4 \][/tex]
Now, substitute the values into the quadratic formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(-6) \pm \sqrt{4}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{6 \pm 2}{2} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{6 + 2}{2} = \frac{8}{2} = 4 \][/tex]
[tex]\[ x_2 = \frac{6 - 2}{2} = \frac{4}{2} = 2 \][/tex]
Hence, the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 - 6x + 8 \)[/tex] are:
[tex]\[ (4, 0) \][/tex]
[tex]\[ (2, 0) \][/tex]
Therefore, the correct [tex]\( x \)[/tex]-intercepts from the given choices are:
[tex]\[ (4, 0) \][/tex]
[tex]\[ (2, 0) \][/tex]
