(2,14),(1,-1),(3,35)
Find the quadratic function y=ax^2+bx+c
1) 4a+2b+c=14
2) a+b+c=-1
3) 9a+3b+c= 35
equation 1&2 = 3a + b = 13
equation 1&3 = -5a +5b = -21
i know you take those two equations now
3a + b = 13
-5a - 5b = -21
multiply the top by 5 and the bottom equation by 3
15a + 5b = 65
-15a - 15b = -63
that
-10b = 2
but its suppose to come out to a whole number what am i doing wrong???
To find the quadratic function y = ax^2 + bx + c, you are given three points (2,14),(1,-1),(3,35).
From the points, you can form three equations:
Equation 1: 4a + 2b + c = 14
Equation 2: a + b + c = -1
Equation 3: 9a + 3b + c = 35
Now, let's solve these equations using the method of elimination/substitution:
First, let's eliminate c:
Multiply Equation 1 by 3 and subtract it from Equation 3 multiplied by 1:
12a + 6b + 3c = 42
-(9a + 3b + c = 35)
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3a + 3b + 2c = 7
This gives us a new equation: 3a + 3b + 2c = 7 ...(Equation 4)
Now, let's eliminate c again:
Multiply Equation 2 by 2 and subtract it from Equation 1 multiplied by 1:
2a + 2b + 2c = -2
-(4a + 2b + c = 14)
-------------------
-2a + c = -16
This gives us a new equation: -2a + c = -16 ...(Equation 5)
Now, we have two equations involving a and b:
Equation 4: 3a + 3b + 2c = 7
Equation 5: -2a + c = -16
Solve Equation 5 for c by adding 2a to both sides:
c = -16 + 2a ...(Equation 6)
Now, substitute Equation 6 into Equation 4:
3a + 3b + 2(-16 + 2a) = 7
3a + 3b - 32 + 4a = 7
7a + 3b - 32 = 7
7a + 3b = 39 ...(Equation 7)
We now have two equations involving a and b:
Equation 7: 7a + 3b = 39
Equation 2: a + b + c = -1
Now, we can solve these two linear equations using the method of elimination/substitution:
Multiply Equation 2 by 7 and subtract it from Equation 7 multiplied by 1:
7a + 3b = 39
-(7a + 7b + 7c = -7)
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-4b - 7c = 46
But we don't need this equation since our objective is to find the quadratic function. We only needed it to eliminate a and solve for c.
Now, we have c = -16 + 2a, which can be simplified to c = 2a - 16.
So, the quadratic function is y = ax^2 + bx + (2a - 16), where a and b are variables.
At this point, you seem to have attempted to solve for b using the eliminated equations.
However, it appears that there is no unique solution for a, b, and c, which results in a whole number. This might mean that there is an error in the given points or equations. Make sure to double-check the points and equations provided to determine if any mistakes were made during the process.