Give an example of a quadratic function which has roots 𝑥=−6
 and 𝑥=9
.

To form a quadratic function from given roots, we can use the fact that if \( x = -6 \) and \( x = 9 \) are roots, then the quadratic function is of the form:

\[
f(x) = a(x - r_1)(x - r_2)
\]

Where \( r_1 \) and \( r_2 \) are the roots. Substituting \( r_1 = -6 \) and \( r_2 = 9 \), we get:

\[
f(x) = a(x + 6)(x - 9)
\]

Now, let's expand the expression:

\[
f(x) = a(x^2 - 9x + 6x - 54)
\]
\[
f(x) = a(x^2 - 3x - 54)
\]

If we choose \( a = 1 \) (the simplest case), the quadratic function becomes:

\[
f(x) = x^2 - 3x - 54
\]

This is a quadratic function with the roots \( x = -6 \) and \( x = 9 \).

Would you like further details on this?

Here are 5 related questions to deepen your understanding:
1. What happens to the function if we change the value of \( a \)?
2. How would you factorize \( f(x) = x^2 - 3x - 54 \)?
3. Can you graph this quadratic function and identify its vertex?
4. How does the discriminant of this function confirm the existence of real roots?
5. How would the function change if one root were imaginary?

**Tip**: The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is \( -\frac{b}{a} \), and the product of the roots is \( \frac{c}{a} \).

Give an example of a quadratic function which has roots 𝑥=−6 and 𝑥=9.

Learn how to create a quadratic function with roots x = -6 and x = 9. This example explains the steps to form the quadratic equation, expand it, and understand related algebraic concepts. Ideal for students in Grades 8-10.

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