Question

34, 19 find two factors of the first number such that their product is the first number and their sum is the second number

Answer 1

To find two factors of 34 that multiply to give 34 and add up to 19, we can denote the two factors as \(x\) and \(y\).

We know:
1. \(x \cdot y = 34\)
2. \(x + y = 19\)

From the second equation, we can express \(y\) in terms of \(x\):
\[ y = 19 - x \]

Now substituting this equation into the first:
\[ x \cdot (19 - x) = 34 \]

Expanding this gives:
\[ 19x - x^2 = 34 \]
Rearranging the equation:
\[ x^2 - 19x + 34 = 0 \]

Now we can solve this quadratic equation using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Where \(a = 1\), \(b = -19\), and \(c = 34\).

Calculating the discriminant:
\[ b^2 - 4ac = (-19)^2 - 4 \cdot 1 \cdot 34 = 361 - 136 = 225 \]

Now substituting into the quadratic formula:
\[ x = \frac{19 \pm \sqrt{225}}{2} = \frac{19 \pm 15}{2} \]

This gives us two potential values for \(x\):
1. \( x = \frac{34}{2} = 17 \)
2. \( x = \frac{4}{2} = 2 \)

Thus, the corresponding values of \(y\) are:
1. If \(x = 17\), then \(y = 19 - 17 = 2\)
2. If \(x = 2\), then \(y = 19 - 2 = 17\)

Therefore, the two factors of 34 that meet the conditions are \(2\) and \(17\).

Verification:
- \(2 \cdot 17 = 34\)
- \(2 + 17 = 19\)

So, the answer is \(2\) and \(17\).