anonymous
  • anonymous
For the given quadratic equation convert into vertex form, find the vertex, and find the value for x = 6. y= -2x^2 + 2x +2?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
@IrishBoy123
anonymous
  • anonymous
@Michele_Laino
Michele_Laino
  • Michele_Laino
if we make this substitution: x=6 into your quadratic equation, we get: \[\Large \begin{gathered} y = - 2 \times {6^2} + 2 \times 6 + 2 = \hfill \\ \hfill \\ = - 2 \times 36 + 12 + 2 = ...? \hfill \\ \end{gathered} \]

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anonymous
  • anonymous
-58
Michele_Laino
  • Michele_Laino
that's right!
anonymous
  • anonymous
So that's the value of x, correct?
Michele_Laino
  • Michele_Laino
that's the value of y, when x=6
anonymous
  • anonymous
So how do we find the value of x and find the vertex?
Michele_Laino
  • Michele_Laino
our task was to find the value of y, when x=6
Michele_Laino
  • Michele_Laino
the general formula of a parabola is: \[\Large y = a{x^2} + bx + c\]
anonymous
  • anonymous
So there is nothing else to the problem it's done that simple lol
Michele_Laino
  • Michele_Laino
we have to find the coordinates of the vertex, and the vertex form of the equation of your parabola
Michele_Laino
  • Michele_Laino
so, as I wrote before, the general equation of a parabola, whose axis is parallel to y axis, is: \[\Large y = a{x^2} + bx + c\] now, please by comparison with the equationof your parabola, what are the coefficients a, b, and c?
anonymous
  • anonymous
6 and -58?
Michele_Laino
  • Michele_Laino
hint: |dw:1435870805196:dw|
anonymous
  • anonymous
ok so its 2,2,and 2
Michele_Laino
  • Michele_Laino
better is a=-2
anonymous
  • anonymous
A=-2, b=2,c=2
Michele_Laino
  • Michele_Laino
that's right!
Michele_Laino
  • Michele_Laino
now the x-coordinate of the vertex is: \[\Large x = - \frac{b}{{2a}} = ...?\]
Michele_Laino
  • Michele_Laino
please substitute your coefficients into taht formula
Michele_Laino
  • Michele_Laino
that*
anonymous
  • anonymous
X=\[-\frac{ 2b }{ 2a}\]
Michele_Laino
  • Michele_Laino
hint: \[\Large x = - \frac{b}{{2a}} = - \frac{2}{{2 \times \left( { - 2} \right)}} = ...?\]
anonymous
  • anonymous
I really dont know
Michele_Laino
  • Michele_Laino
why?
Michele_Laino
  • Michele_Laino
\[\Large x = - \frac{b}{{2a}} = - \frac{2}{{2 \times \left( { - 2} \right)}} = - \frac{2}{{ - 4}} = ...?\]
anonymous
  • anonymous
I Oh ok i thought it was like a trick next step lol
anonymous
  • anonymous
\[\frac{ 1 }{ 2}\]
anonymous
  • anonymous
negative
Michele_Laino
  • Michele_Laino
that's right!
Michele_Laino
  • Michele_Laino
no, it is 1/2
anonymous
  • anonymous
oh ok
Michele_Laino
  • Michele_Laino
next, the y-coordinate of the vertex is given by the subsequent formula: \[\Large y = - \frac{{{b^2} - 4ac}}{{4a}} = ...?\] as before, please substitute your coefficients into that formula
anonymous
  • anonymous
\[y=\frac{ 2^{2}-4ac }{ 4a}\]
Michele_Laino
  • Michele_Laino
hint: \[\Large y = - \frac{{{b^2} - 4ac}}{{4a}} = - \frac{{{2^2} - \left\{ {4 \times \left( { - 2} \right) \times 2} \right\}}}{{4 \times \left( { - 2} \right)}} = ...?\]
anonymous
  • anonymous
\[\frac{ 20 }{ -8}\]
Michele_Laino
  • Michele_Laino
I think: 20/8
Michele_Laino
  • Michele_Laino
am I right?
anonymous
  • anonymous
yes
anonymous
  • anonymous
so its \[2\frac{ 1 }{ 2 }\]
Michele_Laino
  • Michele_Laino
ok! so the vertex V of our parabola is this point: \[\Large V = \left( {\frac{1}{2},\frac{5}{2}} \right)\] since: \[\Large \frac{{20}}{8} = \frac{5}{2}\]
anonymous
  • anonymous
So my answers are -58 for x inputting into the y and my vertex is (1/2) (5/2)? correct
Michele_Laino
  • Michele_Laino
yes! correct!
anonymous
  • anonymous
yay we did it lol
Michele_Laino
  • Michele_Laino
finally, we have to write the vertex form of the equation of your parabola
anonymous
  • anonymous
Thank you very much for your patience
Michele_Laino
  • Michele_Laino
:)
anonymous
  • anonymous
ok im ready lol
Michele_Laino
  • Michele_Laino
first step: we can factor out -2 at the right side of your equation, so we get: \[\Large y = - 2\left( {{x^2} - x - 1} \right)\]
anonymous
  • anonymous
so do we input abc or leave it like that
Michele_Laino
  • Michele_Laino
we have to continue with the second step
Michele_Laino
  • Michele_Laino
here is the second step: \[\Large {x^2} - x - 1 = {x^2} - x + \frac{1}{4} - \frac{1}{4} - 1\]
Michele_Laino
  • Michele_Laino
I added and subtracted 1/4
anonymous
  • anonymous
so wouldnt it just x^2-x-1
Michele_Laino
  • Michele_Laino
yes! I cahnge the form of x^2-x-1 using the rules of algebra
Michele_Laino
  • Michele_Laino
change*
Michele_Laino
  • Michele_Laino
nevertheless the quantity is the same, I change only its algebraic shape
anonymous
  • anonymous
\[x ^{2}-x-1\]
Michele_Laino
  • Michele_Laino
yes that quantity will be the same, I change its form only
Michele_Laino
  • Michele_Laino
next I can simplify that expression above as below: \[\Large \begin{gathered} {x^2} - x - 1 = {x^2} - x + \frac{1}{4} - \frac{1}{4} - 1 = \hfill \\ \hfill \\ = {x^2} - x + \frac{1}{4} - \frac{5}{4} \hfill \\ \end{gathered} \]
Michele_Laino
  • Michele_Laino
now, the quantity: \[\large {x^2} - x + \frac{1}{4}\] is a perfect square, namely it is the square of which binomial?
anonymous
  • anonymous
i have no clue
Michele_Laino
  • Michele_Laino
hint: please compute this: \[\Large {\left( {x - \frac{1}{2}} \right)^2} = ...?\]
anonymous
  • anonymous
|dw:1435879978188:dw|
Michele_Laino
  • Michele_Laino
perfect!
Michele_Laino
  • Michele_Laino
so we can write this: \[\large \begin{gathered} {x^2} - x - 1 = {x^2} - x + \frac{1}{4} - \frac{1}{4} - 1 = \hfill \\ \hfill \\ = {x^2} - x + \frac{1}{4} - \frac{5}{4} = {\left( {x - \frac{1}{2}} \right)^2} - \frac{5}{4} \hfill \\ \end{gathered} \]
Michele_Laino
  • Michele_Laino
namely: \[\large {x^2} - x - 1 = {\left( {x - \frac{1}{2}} \right)^2} - \frac{5}{4}\]
Michele_Laino
  • Michele_Laino
then I substitute that new expression, into the subsequent formula: \[\Large y = - 2\left( {{x^2} - x - 1} \right)\]
Michele_Laino
  • Michele_Laino
and I get: \[\Large y = - 2\left\{ {{{\left( {x - \frac{1}{2}} \right)}^2} - \frac{5}{4}} \right\}\]
anonymous
  • anonymous
\[y=-2\left( x-\frac{ 1 }{ 2 } \right)^{2}+\frac{ 5 }{ 2 }\]
Michele_Laino
  • Michele_Laino
congratulations!! :)
anonymous
  • anonymous
Yay lol
anonymous
  • anonymous
So the vertex is 1/2 and 5/2, that's are equation above and -58 is the input?
Michele_Laino
  • Michele_Laino
yes!
anonymous
  • anonymous
yay thank you so much!
Michele_Laino
  • Michele_Laino
:)

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