Question

3. The solution to the quadratic equation x2 – 11x + 22 = 0 is x = 3 and x = 6. What is the base of the numbers

Answer 1

The solution to the quadratic equation $x^{2} - 11x + 22 = 0$ is $x_{1} = 3$ and $x_{2} = 6$. What is the base of the numbers?

Using Vieta's formulas for quadratic equation:

$$x _ {1} + x _ {2} = 11$$ $$x _ {1} * x _ {2} = 22$$

So

$$(3)_n + (6)_n = (11)_n$$ $$(3)_n * (6)_n = (22)_n$$

Where $n$ - base of the numbers.

$$(3)_n = (3)_{10} = 3$$ $$(6)_n = (6)_{10} = 6$$ $$(11)_n = (n + 1)_{10} = n + 1$$ $$(22)_n = (2n + 2)_{10} = 2n + 2$$

And

$$3 + 6 = n + 1$$ $$3 * 6 = 2n + 2$$

Obviously $n = 8$

Answer: $n = 8$