John B. answered • 08/22/19
BS CHEMISTRY and BS MATHEMATICS TUTOR - NORTH ATLANTA / ONLINE
Completing the square is the best way of solving these for the vertex. When you take Calculus, however, these problems can be solved MUCH more simply.
y = 2x2 + 8x + 7
First, make the a term = 1 by factoring out a 2 as it is necessary if you complete the square:
y = 2(x2 + 4x + 7/2)
Complete the square by adding (b/2)2 and then subtracting it out again:
b = 4, b/2 = 4/2 = 2 (b/2)2 = 4
You need a 4 added inside and with a 2 factored out, so you need to add 8 and subtract 8 back out.
y = 2(x2 + 4x + 7/2) + 4 - 4
Don't forget that the +7/2 has to be multiplied by the factor of 2 which was factored out before.
y = 2(x2 + 4x + 4) + 2*(7/2) - 8
y = 2(x + 2)2 - 1
The vertex is (h,k) with the equation in the form y = a(x-h)2 + k
So the equation is y = 2(x + 2)2 - 1 and the vertex is (-2,-1)
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y = -3x2 + 24x - 31
follow instructions from the other problem as you did before:
y = -3(x2 - 8x + 31/3)
(b/2)2 = (-8/2)2 = 16
But you need the 16 multiplied by the factor of -3 which was factored out already, which is -48
add this and subtract it out and then add it back (since you are subtracting it inside the parentheses and adding it back outside the parentheses.
y = -3(x2 - 8x + 16) + 48
y = -3(x - 4)2 + 48
So the equation is -3(x - 4)2 + 48 and the vertex is (4,48)
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These are the answers that you seek:
The equation is y = 2(x + 2)2 - 1 and the vertex is (-2,-1)
The equation is -3(x - 4)2 + 48 and the vertex is (4,48)
If you like the way I have explained this, please leave a comment, take a look at my tutoring page, and send me a message letting me know you liked it!
Thanks!
John B.
Atlanta, GA
John B.
Actually, this is the completing the square technique. Please note that for the coefficient of a, which is the number in front of the x^2 in the original equation, if this coefficient is not +1, the problem becomes more complicated and you must follow my steps carefully to account for a factor of a taken outside the parentheses. Yesterday, when I wrote this solution, I was lazy and expressed all the x^2 and ( )^2 terms as x2 and ( )2. I can see where that was confusing, since nothing was shown as actually "squared". I apologize if this confused you. Please feel free to leave me another comment and ask specifically about parts of the solutions or anything else that was confusing. Thanks for looking at my solution! John B. Atlanta, GA08/23/19
Sa T.
I like how you did that but in school, we learn it to change the equations to vertex form by completing the square so can you please how it that way?08/23/19