Final answer:
The two numbers that multiply to get 2 and add up to get 4 are ±2 + √2 and 2 - √2, found by forming and solving a quadratic equation with the given sum and product as the coefficients.
Step-by-step explanation:
The problem is to find two numbers that multiply to get 2 and, when added, they equal 4. To find these two numbers, we can set up a system of equations based on the conditions given:
- Let the two numbers be x and y.
- The first condition can be represented as x × y = 2 (they multiply to 2).
- The second condition is represented as x + y = 4 (they add up to 4).
To solve this system, we can use substitution or elimination. An alternative intuitive method is to recognize the conditions resemble the factoring of a quadratic equation's solutions. Since the sum and product of the roots of a quadratic equation ax² + bx + c = 0 are −b/a and c/a respectively, we can form the quadratic equation x² - (sum of numbers)x + (product of numbers) = 0. With the given conditions, this becomes x² - 4x + 2 = 0. Using the quadratic formula to solve for x gives us the two numbers:
x = [-(-4) ± √{{(-4)² - 4 × 1 × 2}}] / (2 × 1)
x = [4 ± √{(16 - 8)}] / 2
x = [4 ± √{8}] / 2
x = [4 ± √{4×2}] / 2
x = [4 ± 2√{2}] / 2
x = 2 ± √{2}
Therefore, the two numbers we are looking for are 2 + √{2} and 2 - √{2}.