Question 263310
standard form of the equation is equal to :


y = ax^2 + bx + c


a,b,c are integers.


when x = 1, y = -8


y = ax^2 + bx + c becomes:


-8 = a*1 + b*1 + c = a + b + c


you get:


a + b + c = -8


when x = 2, y = 5


y = ax^2 + bx + c becomes:


5 = a*2^2 + b*2 + c = a*4 + b*2 + c


you get:


4a + 2b + c = 5


when x = 3, y = 24


y = ax^2 + bx + c becomes:


24 = a*3^2 + b*3 + c


you get:


9a + 3b + c = 24


you now have 3 equations in 3 unknowns that need to be solved simultaneously.


those equations are:


<pre>
a +   b + c = -8
4a + 2b + c =  5
9a + 3b + c = 24
</pre>


solving these equations simultaneously, you get:


a = 3
b = 4
c = -15


to confirm these answers are correct, substitute these values in the original equations to get:


<pre>
a +   b + c = -8
4a + 2b + c =  5
9a + 3b + c = 24
</pre>


becomes:


<pre>
3    +     4      +    (-15)   =   -8
4*3  +     2*4    +    (-15)   =    5
9*3  +     3*4    +    (-15)   =   24
</pre>


simplifying, you get:


<pre>
 3   +     4      +    (-15)   =   -8
12   +     8      +    (-15)   =    5
27   +    12      +    (-15)   =   24
</pre>


combining like terms, you get:


<pre>
     -8   =   -8
      5   =    5
     24   =   24
</pre>


this confirms that the values for a,b,c are good.


plug the values of:


a = 3
b = 4
c = -15


into the equation of y = ax^2 + bx + c to get:


y = 3x^2 + 4x - 15


that's your equation.


take any (x,y) pair of values you are given and they should be confirmed to be true by plugging them into the equation.


for example:


take (x,y) = (7,160) and plug those values into the equation.


y = 3x^2 + 4x - 15 becomes:


160 = 3*(7^2) + 4*7 - 15 which becomes:


160 = 3*49 + 28 - 15 which becomes:


160 = 147 + 28 - 15 which becomes:


160 = 160 which is true confirming the values for a,b,c are good.


I did not show you how I solved the system of 3 equations in 3 unknowns because it would have detracted from the main focus of what I was trying to show you.


If you need help with solving this system of 3 equations in 3 unknowns, let me know and I will show you how it was done.


the question was:


what is the value of a?


the answer is:


the value of a is 3.