$\displaystyle \frac{6x + 13}{3x + 7} = \frac{9x + 4}{12x + 10}$
Put under common denominator.
$\displaystyle \frac{(6x + 13)(12x + 10)}{(3x + 7)(12x + 10)} = \frac{(9x + 4)(3x + 7)}{(12x + 10)(3x + 7)}$
Simplify (get rid of fractions) :
$\displaystyle (6x + 13)(12x + 10) = (9x + 4)(3x + 7)$
Expand :
$\displaystyle 72x^2 + 60x + 156x + 130 = 27x^2 + 63x + 12x + 28$
Simplify :
$\displaystyle 72x^2 + 216x + 130 = 27x^2 + 75x + 28$
Make the right-hand side $\displaystyle 0$ :
$\displaystyle 45x^2 + 141x + 102 = 0$
Use the quadratic formula and you are done :
$\displaystyle \triangle = b^2 - 4ac = 141^2 - 4 \times 45 \times 102 = 1521$
Solutions :
$\displaystyle x_1 = \frac{-b - \sqrt{\triangle}}{2a} = \frac{-141 - 39}{90} = \frac{-180}{90} = -2$
$\displaystyle x_2 = \frac{-b + \sqrt{\triangle}}{2a} = \frac{-141 + 39}{90} = \frac{-102}{90} = \frac{-17}{15}$
Solutions are $\displaystyle -2$ and $\displaystyle \frac{-17}{15}$
Ah, right. But the problem is, coefficients of your equation are quite impressive, why not take smaller numbers so as to get a smaller quadratic equation, easy to understand and factorize mentally (without quadratic formula) for those young people ? Something like $\displaystyle x^2 + 10x - 25$ ? You know, the product and sum of roots : $\displaystyle 25 = 5 \times 5$, but $\displaystyle 10 = 5 + 5$, thus $\displaystyle x^2 + 10x - 25 = (x - 5)(x + 5)$
Test for poster:
$\displaystyle 6x+13+9x+4\longrightarrow15x+17$
$\displaystyle 3x+7+12x+10\longrightarrow15x+17$
$\displaystyle x=\frac{-17}{15}$
$\displaystyle (6x+13)-(3x+7)=3x+6$
$\displaystyle (12x+10)-(9x+4)=3x+6$
$\displaystyle x=\frac{-6}{3}=-2$
Hi Mukilab!
Just saw ur questions on Quadratics.
You are right these can be solved in few seconds with Vedic Math.
Its easy.
I liked ur Poster.When you r through why dont u send it to me ...If you would like I can put it up on our vedic Maths Blog.
Feel free to mail me if you need any more help with regards to Vedic Maths.
Thanks
Warm Wishes
Gaurav
Gaurav Tekriwal
President
The Vedic Math Forum India
www.vedicmathsindia.org
Follow us on Twitter: gtekriwal (VedicMathsIndia) on Twitter
We Blog Actively at : Vedic Maths Forum India Blog
------------------------------------------------------------------------------------
Hello gtekriwal, nice to see it's your first post, and although I am quite a new member, Welcome!
No problem, although I hope you don't mind my caution. I'll hand it in to my maths teacher before I email it to you. Is it OK if I mention this blog to them?
Oh and I'll definitely be hoping to study all the sutras, I only know about half. Hopefully in time my knowledge will gradually expand and I'll learn these much simpler methods. Unfortunately, knowledge comes hand in hand with practice which I have learned in studying vertically by crosswise. It gets fairly confusing with the three digits in your head.
Nice forum by the way! Hope it prospers
~Muki
England in fact, I actually did this as a GnT course (Gifted 'n' Talented) with several other pupils. We had several lessons regarding this and other areas of mathematics and we were asked to create a poster on one or more sutras from vedic maths. I found it a very enjoyable experience!