Question

Find two numbers with sum 4 and product 2.

Hint:

A useful shortcut: a quadratic equation whose roots are the two numbers with a given sum S and product P is z² - Sz + P = 0. Here, S=4 and P=2, so the equation is z² - 4z + 2 = 0, which is the same equation we derived.

Answer 1

We need to find two numbers, let's call them x and y, that satisfy two conditions: their sum is 4 and their product is 2.

We can set up a system of two equations:
1) x + y = 4
2) xy = 2
We can solve this system by substitution, which will lead to a quadratic equation.

From the first equation, we can express y in terms of x:
y = 4 - x Now substitute this expression for y into the second equation:
x(4 - x) = 2 Expand the equation:
4x - x² = 2 Rearrange into the standard quadratic form ax² + bx + c = 0:
x² - 4x + 2 = 0 This equation cannot be easily factored, so we use the quadratic formula to solve for x:
x = -b ± √(b² - 4ac)2a Here, a=1, b=-4, and c=2.
x = -(-4) ± √((-4)² - 4(1)(2))2(1) x = 4 ± √(16 - 8)2 x = 4 ± √(8)2 Simplify the square root: √(8) = √(4 × 2) = 2√(2).
x = 4 ± 2√(2)2 Divide both terms in the numerator by 2:
x = 2 ± √(2) This gives us two possible values for x: x₁ = 2 + √(2) and x₂ = 2 - √(2).
If we take x = 2 + √(2), then y = 4 - x = 4 - (2 + √(2)) = 2 - √(2).
If we take x = 2 - √(2), then y = 4 - x = 4 - (2 - √(2)) = 2 + √(2).
In both cases, the two numbers are the same pair.

The two numbers are 2 + √(2) and 2 - √(2).