Question

If the roots of $x^2 + ax + 9 = 0$ are complex, then

• A. $a<6$
• B. $a<-6$
• C. $|a|<6$
• D. $|a|>6$

Hint:

For quadratic $ax^2+bx+c=0$, roots are complex if $b^2-4ac<0$.

Answer 1

The Correct Option is C

To determine when the roots of the quadratic equation \(x^2 + ax + 9 = 0\) are complex, we need to analyze the discriminant of the equation.

The general form of a quadratic equation is \(ax^2 + bx + c = 0\). For this equation, \(a = 1\), \(b = a\), and \(c = 9\).

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\(\Delta = b^2 - 4ac\)

Substitute the values of \(a\), \(b\), and \(c\) into the formula for the discriminant:

\(\Delta = a^2 - 4 \times 1 \times 9 = a^2 - 36\)

For the roots to be complex, the discriminant must be less than zero:

\(a^2 - 36 < 0\)

Rearranging gives:

\(a^2 < 36\)

Taking the square root of both sides, we get:

\(|a| < 6\)

Therefore, the condition for the roots to be complex is \(|a| < 6\).

Let's evaluate each option:

\(a < 6\)

1. – This is not sufficient as \(|a|\) also includes negative values of \(a\).

\(a < -6\)

1. – This is incorrect because this condition implies \(a\) is only in the negative range below -6.

\(|a| < 6\)

1. – This is correct as it accurately describes the condition determined above.

\(|a| > 6\)

1. – This is incorrect, as it implies that the roots are not complex.

Hence, the answer is \(|a| < 6\).