Find the sum and product of the roots 15x = 3x² - 2

To find the sum and product of the roots for the equation \(15x = 3x^2 - 2\), we first need to rewrite it in the standard quadratic form:

\[
3x^2 - 15x - 2 = 0
\]

Now the equation is in the standard quadratic form \(ax^2 + bx + c = 0\), where:

- \(a = 3\)
- \(b = -15\)
- \(c = -2\)

### Sum and Product of the Roots

For a quadratic equation \(ax^2 + bx + c = 0\), the sum (\(S\)) and product (\(P\)) of its roots can be found using the formulas:

\[
S = -\frac{b}{a}
\]
\[
P = \frac{c}{a}
\]

#### Sum of the Roots

\[
S = -\frac{b}{a} = -\left(-\frac{15}{3}\right) = \frac{15}{3} = 5
\]

#### Product of the Roots

\[
P = \frac{c}{a} = \frac{-2}{3} = -\frac{2}{3}
\]

### Conclusion

- The **sum of the roots** is \(5\).
- The **product of the roots** is \(-\frac{2}{3}\).

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How can you use the quadratic formula to find the exact roots of \(3x^2 - 15x - 2 = 0\)?
2. What does the discriminant tell us about the nature of the roots for the equation \(3x^2 - 15x - 2 = 0\)?
3. How would you factorize the quadratic equation \(3x^2 - 15x - 2 = 0\) to find its roots

Learn how to find the sum and product of roots for the quadratic equation 15x = 3x² - 2. This problem covers the quadratic equation formula, quadratic formula theorem, and provides a detailed step-by-step solution. Suitable for Grades 10-12 students.

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