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

$3x^2 - 2 - 15 = 0$

Simplify this equation:

$3x^2 - 17 = 0$

This is a quadratic equation of the form $ax^2 + bx + c = 0$, where:

• $a = 3$
• $b = 0$
• $c = -17$

Sum and Product of the Roots

For any 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

$S = -\frac{b}{a} = -\frac{0}{3} = 0$

Product of the Roots

$P = \frac{c}{a} = \frac{-17}{3} = -\frac{17}{3}$

Conclusion

• The sum of the roots is $0$.
• The product of the roots is $-\frac{17}{3}$.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the roots of a quadratic equation using the quadratic formula?
2. What is the discriminant of a quadratic equation, and what does it tell us about the roots?
3. How does completing the square work for solving quadratic equations?
4. Can you show how to graph the quadratic equation $3x^2 - 17 = 0$ and interpret the graph?
5. How do the sum and product of roots formulas change if the quadratic equation is written in a different form?

Tip: Remember that the sum and product of the roots formulas apply to any quadratic equation in standard form, making it easier to solve problems without needing to find the roots explicitly.