Get the answer to "Let a, b, c∈N and Quadratic equations x2−2ax+b=0,x2−2bx+c=0,x2−2cx+a=0,,..."

Kendal Day

Answered question

2022-04-14

Let a, b,

c \in N

and Quadratic equations

x^{2} - 2 a x + b = 0, x^{2} - 2 b x + c = 0, x^{2} - 2 c x + a = 0,,

has positive integer roots. Find a, b, c.

star04iks7

Beginner2022-04-15Added 14 answers

Explanation:
Step 1
Consider the discriminants of the quadratic equations. These are:

4 a^{2} - 4 b

4 b^{2} - 4 c

4 c^{2} - 4 a

Or you know what, let's just use the quadratic formula to lay down the roots of x from each equation.

(1) \dots x = \frac{2 a \pm \sqrt{4 a^{2} - 4 b}}{2} = a \pm \sqrt{a^{2} - b}

Similarly for the rest:

(2) \dots x = b \pm \sqrt{b^{2} - c}

(3) \dots x = c \pm \sqrt{c^{2} - a}

Step 2
Discriminants are null

a^{2} - b = 0 \Rightarrow b = a^{2}

b^{2} - c = 0 \Rightarrow c = b^{2} = a^{4}

c^{2} - a = 0 \Rightarrow a^{8} - a = 0

a (a^{7} - 1) = 0 \Rightarrow a = {0, 1}

So either everything is 0 and the equations don't exist or everything is 1 and at least we have something. Hence here's a solution:

{a, b, c} = {1, 1, 1}

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