## $$\textit{Solve : }4x^2-5x-12=0$$

This is a quarratic equation. This is a quadratic equation because the left hand side of the equatin is a polynocial of degree 2.Normally you are advsied to do factorization of the expression in the form of (x-a)(x-b)=0.

This will lead to 2 possible solutions namely x-a=0 i.e. x=a or x-b=0 i.e. y=0.

$$\textit{Unfortunately }, \textit{ it is not possible to factorise } 4x ^ 2 – 5x – 12 \textit{ in the format above.}$$

However, there is good news. There is a method called Sridhar Acharya method after the name of the bengali mathematician.

Sridhar Acharya method provides roots for any quadratic equation of the form:

$$ax^2+bx+c=0\;where\;a.b,c\;are\;real\;and\;a\neq0\textit{and the roots are given by}$$

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

So, if you compare the given equation with Sridhar Acharya equation, a=4. b=-5 and c=-12.

$$So,\;the\;roots\;are\;given\;by\;x=\;\frac{-(-5)\pm\sqrt{{(-5)}^2-4.4.(-12)}}{2.4}$$

$$i.e.\;x=\;\frac{5\pm\sqrt{25+192}}{2.4}$$

$$i.e.\;x=\;\frac{5\pm\sqrt{217}}8$$

$$i.e.\;x=\;\frac{5\pm14.73}8$$

$$i.e.\;the\;2\;solutions\;are\;given\;by\;x=\;\frac{5+14.73}8\;and\;x=\;\frac{5-14.73}8$$

$$i.e.\;x=\;2.6625\;and\;x=1.21625$$

$$Note\;on\;Sridhar\;Acharya\;Method:$$

$$Note\;on\;Sridhar\;Acharya\;Method:\;b^2-4ac\;has\;special\;significance\;and\\ this\;is\;Discriminant\;and \;is\;denoted\;by\;D.$$

$$This\;is\;called\;discriminent\;because\;the\;nature\;of\;the\;roots\;of\;the\;quadratic\\equation\;depends\;on\;the\;value\;of\;D.$$

$$There\;can\;be\;3\;situations:\\1.\;D=0,\;the\;roots\;are\;real\;and\;equal\\2.\;D>0,\;roots\;are\;real\;and\;unequal\\3.\;D<0,\;the\;roots\;are\;imaginary\;and\;unequal\\Since \;D=217\;the\; equation \;has\;2\;real\;unequal\;roots.\;$$