The equation `ax^(2)+6x-9=0` will have imaginary roots if

Options

A. `agt-1`
B. `age-1`
C. `ale1`
D. `-1ltalt1`

Answer

To determine the conditions under which the quadratic equation \( ax^2 + 6x - 9 = 0 \) has imaginary roots, we need to analyze the discriminant of the equation. The discriminant \( D \) is given by the formula:

\[ D = b^2 - 4ac \]

For the equation \( ax^2 + 6x - 9 = 0 \), we identify the coefficients as follows:
- \( a = a \)
- \( b = 6 \)
- \( c = -9 \)

Now, substituting these values into the discriminant formula:

\[ D = 6^2 - 4(a)(-9) \]

Calculating \( D \):

\[ D = 36 + 36a \]

For the roots of the quadratic equation to be imaginary, the discriminant must be less than zero:

\[ D < 0 \]

Substituting the expression for \( D \):

\[ 36 + 36a < 0 \]

Now, we can solve this inequality for \( a \):

1. Subtract 36 from both sides:

\[ 36a < -36 \]

2. Divide both sides by 36:

\[ a < -1 \]

Thus, the condition for the equation \( ax^2 + 6x - 9 = 0 \) to have imaginary roots is:

\[ a < -1 \]

### Summary of Steps:
1. Identify coefficients \( a, b, c \).
2. Calculate the discriminant \( D = b^2 - 4ac \).
3. Set up the inequality \( D < 0 \).
4. Solve for \( a \).