 Write a quadratic function with leading coefficient 1 that has roots of 37 and P. 

To write a quadratic function with roots 37 and \( P \), we use the fact that a quadratic equation with roots \( x_1 \) and \( x_2 \) can be written in factored form as:

\[
f(x) = (x - x_1)(x - x_2)
\]

For your problem, the roots are \( 37 \) and \( P \), so the quadratic function is:

\[
f(x) = (x - 37)(x - P)
\]

Since you want the leading coefficient to be 1, there's no need to multiply by any constant. Expanding this:

\[
f(x) = x^2 - Px - 37x + 37P
\]

\[
f(x) = x^2 - (P + 37)x + 37P
\]

Thus, the quadratic function is:

\[
f(x) = x^2 - (P + 37)x + 37P
\]

Would you like further details or explanations on this process?

Here are five related questions to expand on this:

1. How would the function change if the leading coefficient were not 1?
2. What are the implications of the sum and product of the roots in a quadratic equation?
3. How can you verify if a quadratic equation has certain roots?
4. What does the discriminant tell you about the nature of the roots?
5. How would this quadratic function change if both roots were negative?

**Tip:** The sum of the roots of a quadratic function \( ax^2 + bx + c = 0 \) is given by \( -b/a \), and the product is \( c/a \).

Write a quadratic function with leading coefficient 1 that has roots of 37 and P.

Learn how to write a quadratic function with leading coefficient 1 and roots of 37 and P. This problem demonstrates key algebraic concepts like factored and expanded forms of quadratic equations. Suitable for students in Grades 9-12.

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