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Graphing $y=ax^2+ bx + c$ by completing the square

  1. Add and subtract the square of half the coefficent of $x$.

  2. Group the perfect square trinomial.

  3. Write the trinomial as a square of a binomial.

Rewrite $y = x^2 + 6x + 8$ into $y = a(x-h)^2 + k$.

I've tried solving this but I get a bit confused at the step where I have to "write the trinomial as a square of a binomial". Not exactly sure how to do that.

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Welcome to Math.SE. Thank you for your question. We will be able to better answer it if you give the context of this question, including a more complete quote than the sentence fragment you cite. Also, if you've tried anything so far, please share that as well. –  vadim123 Apr 29 '13 at 2:27
    
"Thank you for your question"? Who thanks for a question asked to him? And perhaps more interesting: why ?! –  DonAntonio Apr 29 '13 at 2:31
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@DonAntonio why not?! –  amWhy Apr 29 '13 at 3:10
    
@amWhy, why why not? We can continue with for ages and it will get pretty boring after a while, and believe me: I know what is "answering" a question with another question... –  DonAntonio Apr 29 '13 at 9:02
    
@SS' : I suggest you change the title of your question, if that is possible. You are trying to graph a quadratic function, which is a little harder than solving the corresponding quadratic equation. –  Stefan Smith Apr 30 '13 at 2:40
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3 Answers

Hint $\rm\,\ \ X^2\! + 2b\, X\! + c\ =\ \overbrace{(X^2\! + 2b\,X\! + b^2)}^{\rm complete\ \ the\ \ \color{#c00}{square}}\! -\! b^2\!+c\ =\ \overbrace{(X + b)^2}^{\rm \color{#c00}{square}} - b^2\!+c $

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$$x^2+6x+8=(x+3)^2-1$$

and now try difference of squares...

In general:

$$x^2\pm ax=\left(x\pm\frac a2\right)^2-\frac{a^2}4$$

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It might help OP to see $x^2+6x+8=x^2+6x+9-9+8=(x+3)^2-1$. Just adding a step. –  Ross Millikan Apr 29 '13 at 2:46
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To complete the square for the equation:

$y = x² + 6x + 8$

Start by:

1) $x^2+6x+8$

2) $x^2+\dfrac 62x +8$ *divide the bx by 2

3) $(x^2+\dfrac 62x + (\dfrac 62)^2)+8$ *square the bx term to form a perfect square

4) $(x^2+ 3x + (36/4))+8$

5) $(x^2+3x + 9)+8-9$ *add the inverse of the c term

6) $(x^2+3x+9)-1$ *factor the perfect square

7) $(x+3)^2-1$

8) $y=(x+3)^2-1$

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