Prove one of the numbers is less than or equal to 4. (1 Viewer)

member 6003 · Jun 3, 2022

Prove that at least one of the numbers 4a-b^2, 4b-c^2, 4c-a^2 is less than or equal to 4.
I don't really know how to approach this, any help would be appreciated.

jimmysmith560 · Jun 3, 2022

Would the following working help?

Assume that all three expressions are greater than 4.

Then we will have $4a-b^2+4b-c^2+4c-a^2>12$

⇔ $a^2-4a+b^2-4b+c^2-4c+12<0$

⇔ $a^2-4a+4+b^2-4b+4+c^2-4c+4<0$

⇔ $\left(a-2\right)^2+\left(b-2\right)^2+\left(c-2\right)^2<0$

member 6003 · Jun 3, 2022

jimmysmith560 said:
Would the following working help?

Assume that all three expressions are greater than 4.

Then we will have $4a-b^2+4b-c^2+4c-a^2>12$

⇔ $a^2-4a+b^2-4b+c^2-4c+12<0$

⇔ $a^2-4a+4+b^2-4b+4+c^2-4c+4<0$

⇔ $\left(a-2\right)^2+\left(b-2\right)^2+\left(c-2\right)^2<0$

thank you

Prove one of the numbers is less than or equal to 4. (1 Viewer)

member 6003

Member

jimmysmith560

Le Phénix Trilingue

member 6003

Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)