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