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I cannot understand how quadratic formula to solve for $x$ was derived.

On this website, it explains the steps

Following I understand

enter image description here

but I cannot understand how they got

$b/2a$

and why they are squaring it

$b^2/4a^2$

Really, I am baffled!

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marked as duplicate by Davide Giraudo, user228113, Mark Bennet, YoTengoUnLCD, user147263 Dec 23 '15 at 19:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ to complete the square $(x+\frac{b}{2a})^2$ $\endgroup$ – janmarqz Dec 23 '15 at 17:30
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Hint:

This is the completing the square trick. Usefull in many other cases and simple to visualize.

enter image description here

The black square is $x^2$, the red rectangle is $bx$, so $x^2+bx$ can be transformed to a square adding the little blue square, that is $(b/2)^2$.

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  • $\begingroup$ Honest to God, the concept still has to soak in. A picture is worth 1000 words. $\endgroup$ – Rhonda Dec 24 '15 at 16:10
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$x^2+\frac{b}{a}x+K^2=(x+K)^2$

$x^2+\frac{b}{a}x+K^2=x^2+2xK+K^2$

$\frac{b}{a}x=2xK$

$k=\frac{b}{2a}$

Then we know that

$x^2+\frac{b}{a}x+(\frac{b}{2a})^2=(x+\frac{b}{2a})^2$

$x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=(x+\frac{b}{2a})^2$

This process is called completing the square. Now we can factor a perfect square. Is this answer sufficient?

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