The Ministry of Road Transport and Highways wants to connect three aspirational districts with two roads 
𝑟
1
r 
1
​
  and 
𝑟
2
r 
2
​
 . Two roads are connected if they intersect. The shape of the two roads 
𝑟
1
r 
1
​
  and 
𝑟
2
r 
2
​
  follows polynomial curve 
𝑓
(
𝑥
)
=
(
𝑥
−
19
)
(
𝑥
−
17
)
2
f(x)=(x−19)(x−17) 
2
  and 
𝑔
(
𝑥
)
=
−
(
𝑥
−
19
)
(
𝑥
−
17
)
g(x)=−(x−19)(x−17) respectively. What will be the 
𝑥
−
x−coordinate of the third aspirational district, if the first two are at 
𝑥
−
x−intercepts of 
𝑓
(
𝑥
)
f(x) and 
𝑔
(
𝑥
)
g(x)

To find the \( x \)-coordinate of the third aspirational district where the two roads intersect, we need to solve for the points of intersection between the two polynomial curves \( f(x) \) and \( g(x) \).

Given:
\[ f(x) = (x - 19)(x - 17)^2 \]
\[ g(x) = - (x - 19)(x - 17) \]

The first two aspirational districts are at the \( x \)-intercepts of \( f(x) \) and \( g(x) \). To find these, we set each function equal to zero.

For \( f(x) = 0 \):
\[ (x - 19)(x - 17)^2 = 0 \]
This gives us the solutions:
\[ x = 19 \]
\[ x = 17 \]

For \( g(x) = 0 \):
\[ - (x - 19)(x - 17) = 0 \]
This also gives us the same solutions:
\[ x = 19 \]
\[ x = 17 \]

Thus, the \( x \)-intercepts of both functions are at \( x = 19 \) and \( x = 17 \).

Next, we need to find the intersection points of \( f(x) \) and \( g(x) \) by setting the two equations equal to each other:
\[ (x - 19)(x - 17)^2 = - (x - 19)(x - 17) \]

For \( x \neq 19 \) (since \( x = 19 \) makes both sides zero and is already an intercept), we can divide both sides by \( (x - 19) \):
\[ (x - 17)^2 = - (x - 17) \]

Rearrange the equation:
\[ (x - 17)^2 + (x - 17) = 0 \]
\[ (x - 17)((x - 17) + 1) = 0 \]
\[ (x - 17)(x - 16) = 0 \]

This gives us the solutions:
\[ x - 17 = 0 \quad \text{or} \quad x - 16 = 0 \]
\[ x = 17 \quad \text{or} \quad x = 16 \]

Since \( x = 17 \) is already known as an intercept, the new solution is:
\[ x = 16 \]

Therefore, the \( x \)-coordinate of the third aspirational district is \( \boxed{16} \).

Would you like more details or have any questions about this solution?

Here are 5 numbered questions you can ask:
1. How do you determine the intersection points of two polynomial functions?
2. Can you explain in detail how to solve polynomial equations?
3. What is the significance of \( x \)-intercepts in polynomial functions?
4. How do polynomial curves generally behave and intersect?
5. Can you provide more examples of solving polynomial intersections?

**Tip:** When solving polynomial equations, always check for common factors to simplify the problem.

Learn how to find the intersection points of polynomial curves to connect aspirational districts. This problem involves polynomial functions and their intersection properties. Suitable for advanced high school students.

Math Problem Statement

The Ministry of Road Transport and Highways wants to connect three aspirational districts with two roads 𝑟 1 r 1 and 𝑟 2 r 2 . Two roads are connected if they intersect. The shape of the two roads 𝑟 1 r 1 and 𝑟 2 r 2 follows polynomial curve 𝑓 ( 𝑥 )

( 𝑥 − 19 ) ( 𝑥 − 17 ) 2 f(x)=(x−19)(x−17) 2 and 𝑔 ( 𝑥 )

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