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:
- \(x \cdot y = 34\)
- \(x + y = 19\)
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 \(x\):
- \( x = \frac{34}{2} = 17 \)
- \( x = \frac{4}{2} = 2 \)
Thus, the corresponding values of \(y\) are:
- If \(x = 17\), then \(y = 19 - 17 = 2\)
- 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 \(17\).