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Answer
To solve the equation \( x^{2} + m x + 9 = 0 \), use the quadratic formula:
\[
x = \frac{-m \pm \sqrt{m^2 - 36}}{2}
\]
This gives two solutions based on the value of \( m \).
Solution
To solve the quadratic equation \( x^{2} + m x + 9 = 0 \), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In this equation, \( a = 1 \), \( b = m \), and \( c = 9 \).
Now, let's calculate the discriminant \( D \) first, which is given by:
\[
D = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
D = m^2 - 4 \cdot 1 \cdot 9 = m^2 - 36
\]
Next, we can substitute \( D \) back into the quadratic formula to find the solutions for \( x \):
\[
x = \frac{-m \pm \sqrt{m^2 - 36}}{2}
\]
Now, let's summarize the solutions:
1. If \( m^2 - 36 > 0 \), there are two distinct real solutions.
2. If \( m^2 - 36 = 0 \), there is one real solution (a repeated root).
3. If \( m^2 - 36 < 0 \), there are no real solutions (the solutions are complex).
Would you like to explore specific values for \( m \) or further analyze the solutions?
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Simplify this solution The Deep Dive
Did you know that quadratic equations like this one date back to ancient civilizations? The Babylonians were solving them around 2000 BC! They had a method that relied on geometric interpretations, which laid the groundwork for all modern algebra. In real-world applications, quadratic equations are everywhere! From determining the trajectory of a ball being thrown to optimizing profits in business by maximizing revenue, these little gems are fundamental in various fields such as physics, engineering, and economics. Next time you see a parabolic curve, remember it's not just a pretty shape—it’s a math problem waiting to be solved!
