The equation in the image is:

$2x^2 + (a - 3)x + 3a - 5 = 0$

You are asked to find $a$ such that the sum and product of the roots of this quadratic equation are equal.

Step 1: General quadratic formula

For any 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 $\frac{C}{A}$.

Step 2: Sum and product of the roots

For the given quadratic equation:

$2x^2 + (a - 3)x + (3a - 5) = 0$

Here, $A = 2$, $B = a - 3$, and $C = 3a - 5$.

• The sum of the roots is $-\frac{B}{A} = -\frac{a - 3}{2}$.
• The product of the roots is $\frac{C}{A} = \frac{3a - 5}{2}$.

Step 3: Set the sum equal to the product

We are told that the sum and product of the roots are equal, so:

$-\frac{a - 3}{2} = \frac{3a - 5}{2}$

Multiply both sides by 2 to eliminate the denominators:

$-(a - 3) = 3a - 5$

Simplify the equation:

$-a + 3 = 3a - 5$

Now solve for $a$:

$3 + 5 = 3a + a$ $8 = 4a$ $a = 2$

Final Answer:

The value of $a$ is $2$.

Would you like further details or clarification?

Here are some additional related questions:

1. How do you find the discriminant of this quadratic equation?
2. What are the specific roots of the equation when $a = 2$?
3. Can the sum and product of roots be negative for certain values of $a$?
4. What happens to the equation when $a = 0$?
5. How does changing the coefficient $A$ affect the sum and product of the roots?

Tip: The relationship between the sum and product of the roots is crucial in solving problems involving quadratic equations.