Solving the Square Equation: ax^2 + bx + c

[Werg22]

I just want to see how can the square equation, $$a(x + {\frac {b} {2a})^{2} + ({c - {\frac {b^2} {4a})$$, can be optained from

$$ax^2 + bx + c$$

Can anyone show me how the equation is manipulated to result into the square form?

 

[Data]

your expression isn't quite right, but the correct one is easy to derive:

$$ax^2+bx+c = a\left( x^2 + \frac{b}{a}x\right) + c = a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right) + c$$

$$= a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) + c - \frac{b^2}{4a} = a\left(x+\frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right).$$

 

[Werg22]

Thank you!

 

[whozum]

Whats the significance fo this?

 

[Werg22]

My math teacher often don't explain the logic of anything and having learned the equation just today I was quite disturbed by it and I wanted to "understand" the equation. That's all. I admit I've been quite silly for not figuring it out...