To determine the nature of the solutions for the quadratic equation \( ax^2 + 8x + c = 0 \), we need to analyze the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
In our equation, \( a \) is the coefficient of \( x^2 \) (which is \( a \)), \( b \) is the coefficient of \( x \) (which is \( 8 \)), and \( c \) is the constant term. Therefore, we can substitute these values into the discriminant formula:
\[ D = 8^2 - 4ac = 64 - 4ac \]
Now, we can determine the nature of the roots based on the value of the discriminant:
-
Two real solutions: This occurs when \( D > 0 \).
\[ 64 - 4ac > 0 \quad \Rightarrow \quad 64 > 4ac \quad \Rightarrow \quad 16 > ac \] -
Imaginary solutions: This occurs when \( D < 0 \).
\[ 64 - 4ac < 0 \quad \Rightarrow \quad 64 < 4ac \quad \Rightarrow \quad 16 < ac \] -
One real solution: This occurs when \( D = 0 \).
\[ 64 - 4ac = 0 \quad \Rightarrow \quad 64 = 4ac \quad \Rightarrow \quad 16 = ac \]
To summarize:
- Two real solutions: \( ac < 16 \)
- Imaginary solutions: \( ac > 16 \)
- One real solution: \( ac = 16 \)