Solve this using the sum & products of the roots -⅚ & -⅙

To solve the problem using the sum and product of the roots, we first identify that the roots of the quadratic equation are given as \(-\frac{5}{6}\) and \(-\frac{1}{6}\).

### Step 1: Recall the Formulas for Sum and Product of Roots
For a quadratic equation of the form:

\[
ax^2 + bx + c = 0
\]

If the roots are \(r_1\) and \(r_2\), the sum and product of the roots can be given by:

\[
r_1 + r_2 = -\frac{b}{a}, \quad r_1 \cdot r_2 = \frac{c}{a}
\]

### Step 2: Find the Sum and Product of the Given Roots
Given roots: 
\[
r_1 = -\frac{5}{6}, \quad r_2 = -\frac{1}{6}
\]

- **Sum of the roots**:

\[
r_1 + r_2 = -\frac{5}{6} + \left(-\frac{1}{6}\right) = -\frac{6}{6} = -1
\]

- **Product of the roots**:

\[
r_1 \cdot r_2 = \left(-\frac{5}{6}\right) \cdot \left(-\frac{1}{6}\right) = \frac{5}{36}
\]

### Step 3: Formulate the Quadratic Equation
Now, use the formulas:

\[
-\frac{b}{a} = -1, \quad \frac{c}{a} = \frac{5}{36}
\]

Let's assume \(a = 36\) to make the calculations easier. Then:

\[
-\frac{b}{36} = -1 \implies b = 36
\]

\[
\frac{c}{36} = \frac{5}{36} \implies c = 5
\]

Thus, the quadratic equation with the given roots is:

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

### Conclusion
The quadratic equation with roots \(-\frac{5}{6}\) and \(-\frac{1}{6}\) is:

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

Would you like a deeper explanation, or do you have any questions?

### Related Questions:
1. How do you derive the quadratic formula using sum and product of roots?
2. What happens to the quadratic equation if the roots are reciprocals of each other?
3. How do changes in coefficients affect the sum and product of roots?
4. Can you derive the quadratic equation from its vertex form?
5. What are the implications of complex roots on the quadratic equation?

**Tip:** Always check if the quadratic equation is factorable when given roots or specific conditions.

Learn how to solve a quadratic equation with roots -5/6 and -1/6 using the sum and product of roots method. This problem covers Vieta's formulas, quadratic equations, and algebraic manipulation. Suitable for high school students.

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