Answer:
The numbers are [tex]8[/tex] and [tex]3[/tex].
Step-by-step explanation:
Let [tex]a[/tex] and [tex]b[/tex] the numbers.
If [tex]a[/tex] and [tex]b[/tex] are factors of [tex]24[/tex], then [tex]ab = 24[/tex]
If [tex]a[/tex] and [tex]b[/tex] has to add up to [tex]11[/tex], the [tex]a +b = 11[/tex]
Solving for [tex]a[/tex] in terms of [tex]b[/tex] in the equation, [tex]a +b = 11[/tex]:
[tex]a +b = 11 \\ a +b -b = 11 -b \\ a = 11 -b[/tex]
Solving the equation [tex]ab = 24[/tex] by plugging in [tex]a[/tex]:
[tex]ab = 24 \\ (11 -b)b = 24 \\ -b^2 +11b = 24 \\ b^2 -11b = -24 \\ b^2 -11b +\frac{121}{4} = -24 +\frac{121}{4} \\ (b -\frac{11}{2})^2 = -\frac{96}{4} +\frac{121}{4} \\ (b -\frac{11}{2})^2 = \frac{25}{4} \\ \sqrt{(b -\frac{11}{2})^2} = \pm \sqrt{\frac{25}{4}} \\ b -\frac{11}{2}= \pm \frac{5}{2}[/tex]
Solving [tex]b[/tex] from the positive root:
[tex]b -\frac{11}{2} +\frac{11}{2} = \frac{5}{2} +\frac{11}{2} \\ b = \frac{16}{2} \\ b = 8[/tex]
Solving [tex]b[/tex] from the negative root:
[tex]b -\frac{11}{2} +\frac{11}{2} = -\frac{5}{2} +\frac{11}{2} \\ b = \frac{6}{2} \\ b = 3[/tex]
Solving for [tex]a[/tex] in the equation, [tex]a = 11 -b[/tex] when [tex]b = 3[/tex]:
[tex]a = 11 -(3) \\ a = 11 -3 \\ a = 8[/tex]
Solving for [tex]a[/tex] in the equation [tex]a = 11 -b[/tex] when [tex]b = 8[/tex]:
[tex]a = 11 -(8) \\ a = 11 -8 \\ a = 3[/tex]
