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For example, $x^2 + 5x + 7$ is $(x + 2.5)^2 + 0.75$ but how would you figure that out? It's useful for proving any quadratic is greater than 0 but it's not always easy to find so. Thanks!

edit: Sorry I'm dumb I didn't see the + 0.75, this is just the vertex form.

ming
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    The magic words are "completing the square". https://en.wikipedia.org/wiki/Completing_the_square – Arturo Magidin Dec 04 '18 at 21:02
  • Yeah, I didn't see the 0.75 so I was confused but it's vertex form isn't it lol – ming Dec 04 '18 at 21:03
  • See also many questions tagged [tag:completing-the-square] –  Dec 04 '18 at 21:05
  • $x^2$ suggest you that teh fist term should be $x$. $5x$ suggest you that this shoud be the $2ab$ term, so it must be $(x+5/2)^2$. Finally, adjust the constant to get the same value. – Tito Eliatron Dec 04 '18 at 21:05
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    Now that you realize this is the vertex form, do you still have a question? – David K Dec 04 '18 at 21:08

1 Answers1

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Yes by completing the square we have that

$$x^2 + 5x + 7=x^2 + 5x + \frac{25}4-\frac{25}4+7=\left(x+\frac52\right)^2+\frac34\ge 0$$

since the two quantities are positive.

The same manipulation is also used for example to obtain the quadratic equation resolution formula and also in the context of quadratic forms.

user
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