To determine whether the quadratic function [tex]\( f(x) = 1.7(x-3)^2 + 3.8 \)[/tex] intersects the [tex]\( x \)[/tex]-axis, we need to find the points where [tex]\( f(x) = 0 \)[/tex].
1. Set the quadratic function equal to zero:
[tex]\[
1.7(x-3)^2 + 3.8 = 0
\][/tex]
2. Isolate the quadratic term:
[tex]\[
1.7(x-3)^2 = -3.8
\][/tex]
3. Divide both sides by 1.7:
[tex]\[
(x-3)^2 = \frac{-3.8}{1.7}
\][/tex]
4. Calculate the quotient [tex]\( \frac{-3.8}{1.7} \)[/tex]:
[tex]\[
(x-3)^2 = -2.235
\][/tex]
5. Analyze the result:
Notice that [tex]\( (x-3)^2 \)[/tex] represents a squared term, which for all real numbers [tex]\( x \)[/tex] is always greater than or equal to zero. However, the quotient [tex]\( -2.235 \)[/tex] is a negative number. It is impossible for a squared term to be equal to a negative number, thus making it impossible to satisfy the equation with any real [tex]\( x \)[/tex] values.
6. Conclusion:
Since [tex]\( (x-3)^2 = -2.235 \)[/tex] has no real solutions, the quadratic function [tex]\( f(x) = 1.7(x-3)^2 + 3.8 \)[/tex] does not intersect the [tex]\( x \)[/tex]-axis at any point.
Therefore, the quadratic function does not intersect the [tex]\( x \)[/tex]-axis at all.
