MHB Proving $1 \leq a \leq 9$ for Quadratic Equations

Albert1
Messages
1,221
Reaction score
0
$a^2-bc-8a+7=0$
$b^2+c^2+bc-6a+6=0$
prove:$1\leq a\leq9$
 
Mathematics news on Phys.org
My solution:

I cheated a bit because my solution is sort of working backwards, since we know we have to prove that $1\le a \le 9$. :o

Since $(a-1)(a-9)=a^2-10a+9$, we manipulate the first equation algebraically to get:

$a^2-10a+9+2a-2-bc=0$

$a^2-10a+9=bc-2(a-1)$(*)

Rewrite the second equation such that it becomes

$b^2+c^2+bc-6a+6=0$

$6(a-1)=(b-c)^2+3bc$

$2(a-1)=\dfrac{(b-c)^2}{3}+bc$

Substitute the above into (*) yields

$a^2-10a+9=bc-\left(\dfrac{(b-c)^2}{3}+bc \right)=-\dfrac{(b-c)^2}{3}\le 0$

$\therefore a^2-10a+9 \le 0$ this shows $1\leq a\leq9$ and we're done.
 
Last edited:
anemone said:
My solution:
I cheated a bit because my solution is sort of working backwards, since we know we have to prove that $1\le a \le 9$. :o Since $(a-1)(a-9)=a^2-10a+9$, we manipulate the first equation algebraically to get: $a^2-10a+9+2a-2-bc=0$ $a^2-10a+9=bc-2(a-1)$(*) Rewrite the second equation such that it becomes $b^2+c^2+bc-6a+6=0$ $6(a-1)=(b-c)^2+3bc$ $2(a-1)=\dfrac{(b-c)^2}{3}+bc$ Substitute the above into (*) yields $a^2-10a+9=bc-\left(\dfrac{(b-c)^2}{3}+bc \right)=-\dfrac{(b-c)^2}{3}\le 0$ $\therefore a^2-10a+9 \le 0$ this shows $1\leq a\leq9$ and we're done.
very good ! working backwards is also a nice method,it gives us a hint to work forwards.
 
Albert said:
$a^2-bc-8a+7=0$
$b^2+c^2+bc-6a+6=0$
prove:$1\leq a\leq9$

we have from 2nd equation
$b^2+c^2+bc = 6a- 6\ \cdots (1)$
from 1st equation
$bc = a^2-8a + 7\ \cdots (2)$
multiply (2) by 3 and subtract from (1)
$(b-c)^2 = 6a-6 - 3(a^2- 8a +7)$
=$-3a^2+30a-27$
=$-3(a^2-10a+9)$
so $a^2-10a + 9 \le 0$
hence $1 \le a \le 9$
 
Last edited:
kaliprasad said:
we have from 2nd equation
$b^2+c^2+bc = 6a- 6\ \cdots (1)$
from 1st equation
$bc = a^2-8a + 7\ \cdots (2)$
multiply (2) by 3 and subtract from (1)
$(b-c)^2 = 6a-6 - 3(a^2- 8a +7)$
=$-3a^2+30a-27$
=$-3(a^2-10a+9)$
so $a^2-10a + 9 \le 0$
hence $1 \le a \le 9$
very nice !
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top