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# Parabola given two equations

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Hi friends,

My stepdaughter came to me today explaining a sum she got in the exam. I believe I have understood her right, so here is the problem....I did some of it..

$$f(x)=a(x+2)^2+c$$

$$g(x)=(2x+5)(x+b)$$

Nothing else is given. Calculate a, b and c. Apparently both equations point to the one and same parabola.

So I said:

The axis of simmetry lies on $$x=-2$$

The x intercepts are $$x=-{5 \over 2}$$ and $$x=-b$$

Now since the one x intercept lies on -2.5, and the axis of simmetry is on the x=-2, it tells me the other x intercept has to be -1.5

So, $$b=1.5$$

Nov 11, 2021

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Ax^2 +4ax+ 4a+c     And

2x^2  +x(5+2b)  +5b       Are supposed to be the same

a=2      5+2b=4a   So b = 3/2       4a+c=5b     So c =    You can finish from here i am sure

Nov 11, 2021
#2
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hi Juriemagic,

If they are the same parabola then they are the same equation.

You need to check the answer,

$$f(x)= a(x^2+4x+4)+c\\ f(x)= ax^2+4ax+(4a+c)\\~\\ g(x)=2x^2+(2b+5)x+5b\\~\\ \text{equate coefficients}\\ a=2\\~\\ 2b+5=4a\\ 2b+5=8\\ 2b=3\\ b=1.5\\ \\~\\ 4a+c=5b\\ 8+c=7.5\\ c=-0.5\\$$

Nov 11, 2021