write the equation in form of y=ax^2+bx+c from the following data: (1,6), (3,26), (-2,21)?

Answer 1

Plug in the points:

For (1, 6) --> x = 1, y = 6
6 = a(1)² + b(1) + c
6 = a + b + c

For (3, 26) --> x = 3, y = 26
26 = a(3)² + b(3) + c
26 = 9a + 3b + c

For (-2, 21) --> x = -2, y = 21
21 = a(-2)² + b(-2) + c
21 = 4a - 2b + c

Now you have the following 3 equations and three unknowns which is straightforward to solve either using substitution or elimination:
6 = a + b + c
26 = 9a + 3b + c
21 = 4a - 2b + c

I like using elimination, but you can use substitution if you like:

Let's subtract equation #1 from equation #2, to eliminate c:
20 = 8a + 2b

Simplify:
10 = 4a + b

Do the same with equation #1 from equation #3, again eliminating c:
15 = 3a - 3b

Simplify:
5 = a - b

Now you have two equations and two unknowns:
10 = 4a + b
5 = a - b

Since they have b and -b, you can just add these equations to eliminate b:
10 + 5 = 4a + a + b - b
15 = 5a
3 = a

Now just plug this back into the other equations to get b and c:
10 = 4a + b
10 = 4(3) + b
10 = 12 + b
-2 = b
b = -2

Finally:
6 = a + b + c
6 = (3) + (-2) + c
6 = 1 + c
c = 5

Final solutions:
a = 3, b = -2, c = 5

So the equation is:
y = 3x² - 2x + 5

As a double-check, I graphed this function and confirmed that the points are indeed on the line.

Answer 2

We know the following

Substitute each point into the equation.

y=ax^2+bx+c

This will give 3 equations in 3 unknowns, a, b and c.

6 = a*1^2 + b*1 + c -----> 6 = a + b + c
26 = a*3^2 + b*3 + c ---> 26 = 9a + 3b + c
21 =a*-2^2 + b*-2 + c -> 21 = 4a - 2b + c

From the 1st equation we know c = 6 - a - b so we substitute that into the 2nd and 3rd equations

26 = 9a + 3b + 6 - a - b
20 = 8a + 2b

21 = 4a - 2b + 6 - a - b
15 = 3a - 3b

Now take 150% of the 1st equation here and add it to the 2nd

20*(1.5) + 15 = (1.5)(8a + 2b) + 3a - 3b
30 + 15 = 12a + 3b + 3a - 3b
45 = 15a

a = 3

Substitute this back into the last equation

15 = 3a - 3b
15 = 3*3 - 3b
15 = 9 - 3b
3b = -6
b = -2

Finally substitute this back into that first constraint we found

c = 6 - a - b
c = 6 - 3 + 2
c = 5

So our equation is

y = 3x^2 - 2x + 5

Answer 3

y= 3x^2 - 2x +5