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

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SUMMARY

The discussion focuses on manipulating the quadratic equation ax² + bx + c into its completed square form, a(x + (b/2a))² + (c - (b²/4a)). The correct derivation involves factoring out 'a' and adding and subtracting (b²/4a²) within the equation. This transformation is essential for solving quadratic equations and understanding their properties, particularly in graphing and finding vertex points. The user expresses a desire for clarity on the logic behind this manipulation, highlighting a common gap in educational explanations.

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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?
 
Last edited:
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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).
 
Thank you!
 
Whats the significance fo this?
 
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...
 
Last edited:

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