To find two numbers that multiply to \( 3636 \) and add up to \( 1212 \), we can set up the following equations:
Let the two numbers be \( a \) and \( b \).
- \( a \times b = 3636 \)
- \( a + b = 1212 \)
To solve this, we can express \( b \) in terms of \( a \) using the second equation:
\[
b = 1212 - a
\]
Now, substitute this expression for \( b \) into the first equation:
\[
a \times (1212 - a) = 3636
\]
Expanding this gives:
\[
1212a - a^2 = 3636
\]
Rearranging the equation:
\[
a^2 - 1212a + 3636 = 0
\]
We can use the quadratic formula to solve for \( a \):
\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -1212 \), and \( c = 3636 \).
Calculating the discriminant:
\[
b^2 - 4ac = (-1212)^2 - 4 \cdot 1 \cdot 3636
\]
\[
= 1460544 - 1454400
\]
\[
= 6144
\]
Now substituting back into the quadratic formula:
\[
a = \frac{1212 \pm \sqrt{6144}}{2}
\]
Calculating \( \sqrt{6144} \):
\[
\sqrt{6144} \approx 78.4
\]
Now, substituting this back in:
\[
a = \frac{1212 \pm 78.4}{2}
\]
Calculating both possible values:
-
Using the plus sign:
\[
a = \frac{1212 + 78.4}{2} \approx \frac{1290.4}{2} \approx 645.2
\]
-
Using the minus sign:
\[
a = \frac{1212 - 78.4}{2} \approx \frac{1133.6}{2} \approx 566.8
\]
Now let's find \( b \):
If \( a \approx 645.2 \), then
\[
b \approx 1212 - 645.2 \approx 566.8
\]
Therefore, the two numbers that multiply to \( 3636 \) and add up to \( 1212 \) are approximately:
\[
\boxed{645.2 \text{ and } 566.8}
\]
However, it seems that since these are not integers, it is best to find precise integers. Let's find the two numbers through factor pairs of \( 3636 \) since they must be whole numbers.
The pairs of factors of \( 3636 \) include \( (1, 3636), (2, 1818), (3, 1212), (6, 606), (9, 404), (18, 202), (36, 101), (2, 1818) \), and we look for ones that add to \( 1212 \):
Testing pairs of factors:
- \( 606 + 606 = 1212 \) and \( 606 \times 6 = 3636\).
So the exact numbers you're looking for are:
\[
\boxed{606 \text{ and } 606}
\]
1 answer