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 quadratic equation \( 8x^2 - 6x - a - 3 = 0 \) given that one root is the square of the other, we can follow these steps:

### Step 1: Define the roots
Let the roots of the equation be \( r \) and \( r^2 \), where \( r \) is one root and \( r^2 \) is the square of the other root.

### Step 2: Use Vieta's formulas
According to Vieta's formulas for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( r + r^2 = -\frac{b}{a} \)
- The product of the roots \( r \cdot r^2 = \frac{c}{a} \)

For our equation \( 8x^2 - 6x - (a + 3) = 0 \):
- Here, \( a = 8 \), \( b = -6 \), and \( c = -(a + 3) \).

### Step 3: Set up the equations
From Vieta's formulas:
1. \( r + r^2 = -\frac{-6}{8} = \frac{3}{4} \) (Sum of roots)
2. \( r \cdot r^2 = \frac{-(a + 3)}{8} \) (Product of roots)

### Step 4: Solve for \( r \)
From the first equation, we can express \( r^2 \) in terms of \( r \):
\[ r^2 + r - \frac{3}{4} = 0 \]
This is a quadratic equation in \( r \). We can solve it using the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot \left(-\frac{3}{4}\right)}}{2 \cdot 1} \]
\[ = \frac{-1 \pm \sqrt{1 + 3}}{2} = \frac{-1 \pm 2}{2} \]
Thus, we have two possible values for \( r \):
1. \( r = \frac{1}{2} \)
2. \( r = -\frac{3}{2} \)

### Step 5: Find corresponding values of \( a \)
Now, we can find the corresponding values of \( a \) using the product of the roots:
1. For \( r = \frac{1}{2} \):
\[ r \cdot r^2 = \frac{1}{2} \cdot \left(\frac{1}{2}\right)^2 = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \]
Setting this equal to the product of the roots:
\[ \frac{1}{8} = \frac{-(a + 3)}{8} \]
This simplifies to:
\[ 1 = -(a + 3) \implies a + 3 = -1 \implies a = -4 \]

2. For \( r = -\frac{3}{2} \):
\[ r \cdot r^2 = -\frac{3}{2} \cdot \left(-\frac{3}{2}\right)^2 = -\frac{3}{2} \cdot \frac{9}{4} = -\frac{27}{8} \]
Setting this equal to the product of the roots:
\[ -\frac{27}{8} = \frac{-(a + 3)}{8} \]
This simplifies to:
\[ -27 = -(a + 3) \implies a + 3 = 27 \implies a = 24 \]

### Final Answer
The values of \( a \) are \( -4 \) and \( 24 \).