https://file.aiquickdraw.com/mathbot/file/1726976986183yk9xw5dt.jpg

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.

Find a so that the sum and product of the roots of the equation 2x^2 + (a - 3)x + 3a - 5 = 0 are equal.

Learn how to solve a quadratic equation where the sum and product of the roots are equal. The equation is 2x^2 + (a - 3)x + 3a - 5 = 0. This guide covers algebraic concepts, including the sum and product of the roots, and is suitable for students in Grades 9-11.

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