If one root of the equation` 8x^(2) - 6x-a-3=0` is the square of the other values of a are :

Options

A. `4,-24`
B. `4,24`
C. `-4,-24`
D. `-4,24`

Answer

To solve the problem, we need to find the values of \( a \) in the quadratic equation \( 8x^2 - 6x - a - 3 = 0 \) given that one root is the square of the other. Let's denote the roots of the equation as \( \alpha \) and \( \beta \), where we know that \( \beta = \alpha^2 \).

### Step 1: Use the sum of the roots
According to Vieta's formulas, the sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{-6}{8} = \frac{6}{8} = \frac{3}{4} \]
Substituting \( \beta = \alpha^2 \) into the equation gives:
\[ \alpha + \alpha^2 = \frac{3}{4} \]

### Step 2: Rearranging the equation
Rearranging the equation, we have:
\[ \alpha^2 + \alpha - \frac{3}{4} = 0 \]
To eliminate the fraction, multiply through by 4:
\[ 4\alpha^2 + 4\alpha - 3 = 0 \]

### Step 3: Factor the quadratic
Now we can factor the quadratic:
\[ 4\alpha^2 + 6\alpha - 2\alpha - 3 = 0 \]
Grouping the terms gives:
\[ (4\alpha^2 + 6\alpha) + (-2\alpha - 3) = 0 \]
Factoring out common terms:
\[ 2\alpha(2\alpha + 3) - 1(2\alpha + 3) = 0 \]
This can be factored as:
\[ (2\alpha + 3)(2\alpha - 1) = 0 \]

### Step 4: Solve for \( \alpha \)
Setting each factor to zero gives:
1. \( 2\alpha + 3 = 0 \) → \( \alpha = -\frac{3}{2} \)
2. \( 2\alpha - 1 = 0 \) → \( \alpha = \frac{1}{2} \)

### Step 5: Find corresponding values of \( a \)
Using the product of the roots, we know:
\[ \alpha \cdot \beta = -\frac{\text{constant term}}{\text{coefficient of } x^2} = -\frac{-a - 3}{8} = \frac{a + 3}{8} \]
Since \( \beta = \alpha^2 \), we have:
\[ \alpha \cdot \alpha^2 = \alpha^3 \]
Thus:
\[ \alpha^3 = \frac{a + 3}{8} \]
Now we can find \( a \) for both values of \( \alpha \).

#### For \( \alpha = \frac{1}{2} \):
\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
So:
\[ \frac{1}{8} = \frac{a + 3}{8} \]
Multiplying through by 8:
\[ 1 = a + 3 \quad \Rightarrow \quad a = 1 - 3 = -2 \]

#### For \( \alpha = -\frac{3}{2} \):
\[ \left(-\frac{3}{2}\right)^3 = -\frac{27}{8} \]
So:
\[ -\frac{27}{8} = \frac{a + 3}{8} \]
Multiplying through by 8:
\[ -27 = a + 3 \quad \Rightarrow \quad a = -27 - 3 = -30 \]

### Final Answer
The values of \( a \) are \( -2 \) and \( -30 \).