MHB What are the real numbers a, b, and c that satisfy certain conditions?

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Determine all three real numbers $a, \,b$ and $c$ which satisfies the conditions $a^2+ b^2+ c^2= 26$, $a + b = 5$ and $b + c\ge 7.$
 
Mathematics news on Phys.org
My attempt:

$a^2+b^2+c^2 = 26\;\;\;\;(1).$

$a+b = 5\;\;\;\;(2).$

$b+c \ge 7\;\;\;\;(3).$

Combining $(1)$ and $(2)$:

\[a^2+b^2+c^2 = (5-b)^2+b^2+c^2 = 26\Rightarrow 2b^2-10b+c^2-1 = 0, \;\;\;\;(4).\]

From $(3)$ one gets: \[c^2 \geq (7-b)^2, \;\;\;\; (5).\]

Combining $(4)$ and $(5)$:

\[2b^2-10b+c^2-1 = 0 \geq 2b^2-10b+(7-b)^2-1 = 3b^2-24b+48\]

\[\Rightarrow b^2-8b+16 \leq 0\]

Or \[(b-4)^2 \leq 0\]

So the only possible $b$-value is $4$. Thus $a = 1$ (from $(2)$) and $c = 3$ (from $(1)$ and $(3)$).
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...

Similar threads

Replies
7
Views
2K
Replies
2
Views
1K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
1
Views
856
Replies
1
Views
1K
Replies
13
Views
2K
Replies
1
Views
1K
Back
Top