To form a quadratic equation with the roots \( \frac{5}{6} \) and \( \frac{1}{6} \), we can use the fact that a quadratic equation can be written in the form:

\[
(x - \text{root}_1)(x - \text{root}_2) = 0
\]

In this case, the roots are \( \frac{5}{6} \) and \( \frac{1}{6} \). Substituting these into the general form:

\[
(x - \frac{5}{6})(x - \frac{1}{6}) = 0
\]

Now, we'll expand this expression:

\[
\left( x - \frac{5}{6} \right) \left( x - \frac{1}{6} \right) = 0
\]

Using the distributive property (FOIL method):

\[
x^2 - \frac{1}{6}x - \frac{5}{6}x + \frac{5}{36} = 0
\]

Simplify the terms:

\[
x^2 - \frac{6}{6}x + \frac{5}{36} = 0
\]

This simplifies to:

\[
x^2 - x + \frac{5}{36} = 0
\]

To eliminate the fraction, multiply the entire equation by 36:

\[
36x^2 - 36x + 5 = 0
\]

So, the quadratic equation with the roots \( \frac{5}{6} \) and \( \frac{1}{6} \) is:

\[
36x^2 - 36x + 5 = 0
\]

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

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Here are 5 related questions for further exploration:
1. How do you find the sum and product of the roots of a quadratic equation?
2. How can you convert a quadratic equation from standard form to factored form?
3. What is the discriminant, and how can it help determine the nature of the roots?
4. How do you solve a quadratic equation using the quadratic formula?
5. How can you graph the quadratic equation and identify its vertex and axis of symmetry?

**Tip:** Always check for common factors before expanding or simplifying an equation—it can save time and reduce mistakes.

Explore how to derive the quadratic equation with roots 5/6 and 1/6. Learn key algebraic techniques including factoring and simplifying quadratic expressions, along with step-by-step expansion using the FOIL method. Suitable for Grades 9-10.

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