To determine which equation has solutions [tex]\( m = -5 \)[/tex] and [tex]\( m = 9 \)[/tex], let's examine each option step-by-step.
### Option 1: [tex]\((m + 5)(m - 9) = 0\)[/tex]
To find the solutions of this equation, set both factors equal to zero:
[tex]\[
(m + 5) = 0 \quad \text{or} \quad (m - 9) = 0
\][/tex]
Solving these equations:
[tex]\[
m = -5 \quad \text{or} \quad m = 9
\][/tex]
This matches the given solutions [tex]\( m=-5 \)[/tex] and [tex]\( m=9 \)[/tex].
### Option 2: [tex]\((m - 5)(m + 9) = 0\)[/tex]
To find the solutions of this equation, set both factors equal to zero:
[tex]\[
(m - 5) = 0 \quad \text{or} \quad (m + 9) = 0
\][/tex]
Solving these equations:
[tex]\[
m = 5 \quad \text{or} \quad m = -9
\][/tex]
These solutions do not match the given solutions [tex]\( m=-5 \)[/tex] and [tex]\( m=9 \)[/tex].
### Option 3: [tex]\(m^2 - 5m + 9 = 0\)[/tex]
We would need to solve this quadratic equation to find its roots using the quadratic formula:
[tex]\[
m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 9 \)[/tex]. Plugging into the quadratic formula:
[tex]\[
m = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(9)}}{2(1)}
\][/tex]
[tex]\[
m = \frac{5 \pm \sqrt{25 - 36}}{2}
\][/tex]
[tex]\[
m = \frac{5 \pm \sqrt{-11}}{2}
\][/tex]
The discriminant [tex]\( \sqrt{-11} \)[/tex] is not a real number, so this equation does not have real roots. The solutions are complex, not matching the given real solutions [tex]\( m=-5 \)[/tex] and [tex]\( m=9 \)[/tex].
### Option 4: [tex]\(m^2 + 5m - 9 = 0\)[/tex]
Similarly, solve this quadratic equation using the quadratic formula:
[tex]\[
m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -9 \)[/tex]. Plugging into the quadratic formula:
[tex]\[
m = \frac{-5 \pm \sqrt{5^2 - 4(1)(-9)}}{2(1)}
\][/tex]
[tex]\[
m = \frac{-5 \pm \sqrt{25 + 36}}{2}
\][/tex]
[tex]\[
m = \frac{-5 \pm \sqrt{61}}{2}
\][/tex]
The roots involve [tex]\(\sqrt{61}\)[/tex], which are not the given simple integer solutions [tex]\( m=-5 \)[/tex] and [tex]\( m=9 \)[/tex].
### Conclusion
The correct equation that has solutions [tex]\( m = -5 \)[/tex] and [tex]\( m = 9 \)[/tex] is:
\[
(m + 5)(m - 9) = 0
