To determine the number of real solutions for the quadratic equation [tex]\( 8x^2 - 11x - 3 = 0 \)[/tex], we follow these steps:
### Step 1: Identify the coefficients
For the quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], we identify the coefficients:
- [tex]\( a = 8 \)[/tex]
- [tex]\( b = -11 \)[/tex]
- [tex]\( c = -3 \)[/tex]
### Step 2: Calculate the discriminant
The discriminant ([tex]\( \Delta \)[/tex]) of a quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
### Step 3: Plug in the coefficients into the discriminant formula
Substitute [tex]\( a = 8 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = -3 \)[/tex] into the formula:
[tex]\[ \Delta = (-11)^2 - 4 \cdot 8 \cdot (-3) \][/tex]
### Step 4: Calculate the value of the discriminant
[tex]\[ \Delta = 121 - (-96) \][/tex]
[tex]\[ \Delta = 121 + 96 \][/tex]
[tex]\[ \Delta = 217 \][/tex]
### Step 5: Determine the number of real solutions based on the discriminant
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real solutions.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly one real solution.
- If [tex]\( \Delta < 0 \)[/tex], there are no real solutions (the solutions are complex).
Since the discriminant [tex]\( \Delta = 217 \)[/tex], which is greater than 0, this equation has two real solutions.
### Final Answer:
The quadratic equation [tex]\( 8x^2 - 11x - 3 = 0 \)[/tex] has two real solutions.
Therefore, the correct answer is:
b. two real solutions
