# Maths Question

• Feb 3rd 2008, 11:17 PM
acevipa
Maths Question
I'm having trouble working out this question. Could someone please provide working on how to solve this.

If $\displaystyle 2x = a + b + c$, show that $\displaystyle (x-a)^2 + (x-b)^2 + (x-c)^2 + x^2 = a^2 + b^2 + c^2$
• Feb 3rd 2008, 11:46 PM
mr fantastic
Quote:

Originally Posted by acevipa
I'm having trouble working out this question. Could someone please provide working on how to solve this.

If $\displaystyle 2x = a + b + c$, show that $\displaystyle (x-a)^2 + (x-b)^2 + (x-c)^2 + x^2 = a^2 + b^2 + c^2$

There are elegant ways and brutal ways ......

Brutal:

1. Sub $\displaystyle x = \frac{a}{2} + \frac{b}{2} + \frac{c}{2}$ into $\displaystyle (x-a)^2 + (x-b)^2 + (x-c)^2 + x^2$.

2. Expand the above.

3. Simplify the above.
• Feb 4th 2008, 12:20 AM
mr fantastic
Quote:

Originally Posted by acevipa
I'm having trouble working out this question. Could someone please provide working on how to solve this.

If $\displaystyle 2x = a + b + c$, show that $\displaystyle (x-a)^2 + (x-b)^2 + (x-c)^2 + x^2 = a^2 + b^2 + c^2$

And elegant:

$\displaystyle (x-a)^2 + (x-b)^2 + (x-c)^2 + x^2 = x^2 - 2ax + a^2 + x^2 - 2bx + b^2 + x^2 - 2cx + c^2 + x^2$

$\displaystyle = a^2 + b^2 + c^2 + 4x^2 -2x(a + b + c)$.

Substitute $\displaystyle a + b + c = 2x$:

$\displaystyle = a^2 + b^2 + c^2 + 4x^2 -4x^2 = a^2 + b^2 + c^2$.

Let your mood decide which way is for you.