Please help to solve Quadratic Function Equation

[vickydera07]

Hello there! I am new to this site and I was wondering if I could have some help in solving this question. It comes from a senior Maths textbook and is part of the simultaneous equations section.
I managed to form three equations using the three points and solved them to find the value of a but I'm stuck at this point and don't know how I would solve to find x after finding b and c.
Roughwork is attached below if anyone is curious to view my solving process. Apologies if the quality is not too great; my writing can be a bit messy. Any feedback is appreciated! Thanks.

 

[Otis]

I managed to … find the value of a

… my writing can be a bit messy

Hi Vicky. Thanks for sharing your work! I think your handwriting is clean and your steps are easy to follow. You did a good job so far using the elimination method, but there's a minor sign error that led to an incorrect value for a.

Your result for equation (D) is correct: 2a – c = 0

Next, it appears that you intended to eliminate c, by multiplying -1 × (D) and adding the result to (E). But, you wrote the factor +1 instead of -1. Maybe that caused you to change the sign of -c only (forgetting about 2a).

Here's a Tip: When we have opposite terms (like 6b and -6b), we eliminate by adding the equations. When we have terms that are identical (like -c and -c), we may eliminate by subtracting either equation from the other. In other words, you could subtract (D) from (E) without having to multiply anything:

(E) 6a – c = 4
(D) 2a – c = 0
---------------

After correcting a, I suspect that you know how to back-substitute to find c and b. If you do need help with that, please let us know.

don't know how I would solve to find x after finding b and c

The good news is that you don't need to find any values for x. They've asked you to "solve to find f(x)". Remember, symbol f(x) is just another way of writing variable y (we call it 'function notation'). So, the 'solve' part refers to getting the coefficient values a,b,c. The 'find f(x)' part refers simply to writing the specific quadratic polynomial which defines function f (the function whose parabolic graph passes through the given points).

That is, report your answer in the form:

f(x) = ax^2 + bx + c

by substituting your values for the coefficients a,b,c.

It's a good idea to check your answer, using the three given (x,y) coordinates. Confirm that each leads to a true equation.
[imath]\;[/imath]

 

[jonah2.0]

Beer inspired diagram follows.

Hello there! I am new to this site and I was wondering if I could have some help in solving this question. It comes from a senior Maths textbook and is part of the simultaneous equations section.
I managed to form three equations using the three points and solved them to find the value of a but I'm stuck at this point and don't know how I would solve to find x after finding b and c.
Roughwork is attached below if anyone is curious to view my solving process. Apologies if the quality is not too great; my writing can be a bit messy. Any feedback is appreciated! Thanks. View attachment 38240
View attachment 38239

 

[vickydera07]

Hi Vicky. Thanks for sharing your work! I think your handwriting is clean and your steps are easy to follow. You did a good job so far using the elimination method, but there's a minor sign error that led to an incorrect value for a.

Your result for equation (D) is correct: 2a – c = 0

Next, it appears that you intended to eliminate c, by multiplying -1 × (D) and adding the result to (E). But, you wrote the factor +1 instead of -1. Maybe that caused you to change the sign of -c only (forgetting about 2a).

Here's a Tip: When we have opposite terms (like 6b and -6b), we eliminate by adding the equations. When we have terms that are identical (like -c and -c), we may eliminate by subtracting either equation from the other. In other words, you could subtract (D) from (E) without having to multiply anything:

(E) 6a – c = 4
(D) 2a – c = 0
---------------

After correcting a, I suspect that you know how to back-substitute to find c and b. If you do need help with that, please let us know.


The good news is that you don't need to find any values for x. They've asked you to "solve to find f(x)". Remember, symbol f(x) is just another way of writing variable y (we call it 'function notation'). So, the 'solve' part refers to getting the coefficient values a,b,c. The 'find f(x)' part refers simply to writing the specific quadratic polynomial which defines function f (the function whose parabolic graph passes through the given points).

That is, report your answer in the form:

f(x) = ax^2 + bx + c

by substituting your values for the coefficients a,b,c.

It's a good idea to check your answer, using the three given (x,y) coordinates. Confirm that each leads to a true equation.
[imath]\;[/imath]

Thanks so much for responding to my question Otis! Your explanation was super clear and I understand where I went wrong now. I often have trouble with signs and end up mixing them up way too much . I didn't really understand how to exactly eliminate the c terms in (D) and (E) since were both equal but after doing some research I finally got it.
I've corrected my answer and found the values of a,b,c which are (a,b,c) = (1, -1, 2) ; therefore:
f(x) = x^2 - x + 2
Once again, many thanks for helping me! I appreciate it a lot!