Solving for x in terms of y: Methods and Alternatives

[phymatter]

Can Anyone Understand this?

See the attachment///
Can Anyone Understand this?
I mean what metod has he followed to solve this equation for x in terms of y , does someone have any method to solve it other than the quadratic formula?/

 

[phyzguy]

It's called "completing the square", and it's where the quadratic formula comes from. Remember, if I have:
x^2+2ax + a^2 = 0 , I can write it as
(x+a)^2 = 0 .
So if I have
ax^2+bx+c = 0
I can write it as:
x^2+\frac{b}{a}x+(\frac{b}{2a})^2 =(\frac{b}{2a})^2-\frac{c}{a}
where I have added (\frac{b}{2a})^2 to both sides. This is also:
(x+\frac{b}{2a})^2 = (\frac{b}{2a})^2-\frac{c}{a}
or:
(x+\frac{b}{2a}) =\pm\sqrt{(\frac{b}{2a})^2-\frac{c}{a}}
or:
x =-\frac{b}{2a}\pm\sqrt{(\frac{b}{2a})^2-\frac{c}{a}}
or:
x =\frac{-b\pm\sqrt{b^2-4ac}}{2a}

This is what they have done, but with \frac{7y+13}{24} playing the role of \frac{b}{2a}

 

[phymatter]

Thanks friend