If ` ( x - 3) and ( 2 x - 5)` are factors of the equation ` 2 x ^(2) + k_(1) x + k_(2) = ` 0, then the values of ` k_(1) and k_(2)` are

Options

A. `k_(1) = 11, k_(2) = - 15`
B. ` k_(1) = - 11, k_(2) = - 15`
C. ` k_(1) = - 11 , k_(2) = 15`
D. ` k_(1) = 11, k_(2) = 15`

Answer

To find the values of \( k_1 \) and \( k_2 \) in the equation \( 2x^2 + k_1 x + k_2 = 0 \), given that \( (x - 3) \) and \( (2x - 5) \) are factors, we can follow these steps:

### Step 1: Write the equation in factored form
Since \( (x - 3) \) and \( (2x - 5) \) are factors of the quadratic equation, we can express the equation as:
\[ (x - 3)(2x - 5) = 0 \]

### Step 2: Expand the factors
Now, we will expand the expression \( (x - 3)(2x - 5) \):
\[ (x - 3)(2x - 5) = x \cdot 2x + x \cdot (-5) - 3 \cdot 2x - 3 \cdot (-5) \]
This simplifies to:
\[ = 2x^2 - 5x - 6x + 15 \]

### Step 3: Combine like terms
Now, we combine the like terms:
\[ 2x^2 - 5x - 6x + 15 = 2x^2 - 11x + 15 \]

### Step 4: Compare with the standard form
Now we compare this with the standard form of the quadratic equation \( 2x^2 + k_1 x + k_2 = 0 \):
\[ 2x^2 - 11x + 15 = 2x^2 + k_1 x + k_2 \]
From this comparison, we can identify:
\[ k_1 = -11 \quad \text{and} \quad k_2 = 15 \]

### Final Answer
Thus, the values of \( k_1 \) and \( k_2 \) are:
\[ k_1 = -11, \quad k_2 = 15 \]

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