Question

Two factors of 24 add up to 11. What are they?​

Answer 1

ANSWER

My answer is in the photo above

Answer 2

Answer:

The numbers are $8$ and $3$.

Step-by-step explanation:

Let $a$ and $b$ the numbers.

If $a$ and $b$ are factors of $24$, then $ab = 24$

If $a$ and $b$ has to add up to $11$, the $a +b = 11$

Solving for $a$ in terms of $b$ in the equation, $a +b = 11$:

$a +b = 11 \\ a +b -b = 11 -b \\ a = 11 -b$

Solving the equation $ab = 24$ by plugging in $a$:

$ab = 24 \\ (11 -b)b = 24 \\ -b^2 +11b = 24 \\ b^2 -11b = -24 \\ b^2 -11b +\frac{121}{4} = -24 +\frac{121}{4} \\ (b -\frac{11}{2})^2 = -\frac{96}{4} +\frac{121}{4} \\ (b -\frac{11}{2})^2 = \frac{25}{4} \\ \sqrt{(b -\frac{11}{2})^2} = \pm \sqrt{\frac{25}{4}} \\ b -\frac{11}{2}= \pm \frac{5}{2}$

Solving $b$ from the positive root:

$b -\frac{11}{2} +\frac{11}{2} = \frac{5}{2} +\frac{11}{2} \\ b = \frac{16}{2} \\ b = 8$

Solving $b$ from the negative root:

$b -\frac{11}{2} +\frac{11}{2} = -\frac{5}{2} +\frac{11}{2} \\ b = \frac{6}{2} \\ b = 3$

Solving for $a$ in the equation, $a = 11 -b$ when $b = 3$:

$a = 11 -(3) \\ a = 11 -3 \\ a = 8$

Solving for $a$ in the equation $a = 11 -b$ when $b = 8$:

$a = 11 -(8) \\ a = 11 -8 \\ a = 3$