To find the sum and product of the roots of the quadratic equation $2x^2 + 3 = 6x$, let's first rewrite the equation in standard form:

$2x^2 - 6x + 3 = 0$

Now, for a quadratic equation of the form $ax^2 + bx + c = 0$:

• The sum of the roots is given by $-\frac{b}{a}$.
• The product of the roots is given by $\frac{c}{a}$.

For the equation $2x^2 - 6x + 3 = 0$:

• $a = 2$
• $b = -6$
• $c = 3$

Sum of the roots:

$\text{Sum} = -\frac{b}{a} = -\frac{-6}{2} = \frac{6}{2} = 3$

Product of the roots:

$\text{Product} = \frac{c}{a} = \frac{3}{2}$

Final Answer:

• The sum of the roots is $3$.
• The product of the roots is $\frac{3}{2}$.

Would you like more details on how to solve this or any related topic?

Here are five related questions you might find interesting:

1. How do you find the roots of the quadratic equation $2x^2 - 6x + 3 = 0$?
2. What is the discriminant of the quadratic equation $2x^2 - 6x + 3 = 0$ and what does it tell us about the roots?
3. How can you derive the quadratic formula for solving any quadratic equation?
4. What are some practical applications of finding the sum and product of roots in real-world problems?
5. How does the graph of the equation $2x^2 - 6x + 3 = 0$ look like and what does it tell us about the roots?

Tip: Understanding the relationship between the coefficients and the roots of a quadratic equation can help you quickly find important properties of the equation without solving it completely.