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Question 263310: The table shows the relationship between x and y in a quadratic equation of the form y = ax^2 + bx + c, where a, b, and c are integers.What is the value of a?
x 1 2 3 4 5 6 7
y -8 5 24 49 80 117 160
Found 2 solutions by richwmiller, Theo:
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! x=1 y=-8
ax^2+bx+c=y
c = -a-b-8, x = 1, y = -8
c = -4 a-2 b+5, x = 2, y = 5
c = -3 (3 a+b-8), x = 3, y = 24
c = -16 a-4 b+49, x = 4, y = 49
c = -a-b-8 and
c = -4a-2 b+5 and
c = -3*(3a+b-8) and
c = -16a-4b+49
a = 3, b = 4, c = -15
y=3x^2+4x-15
y=(x+3)(3x-5)
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,3,5,9,15,45
-1,-3,-5,-9,-15,-45
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-45) = -45
3*(-15) = -45
5*(-9) = -45
(-1)*(45) = -45
(-3)*(15) = -45
(-5)*(9) = -45
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | -45 | 1+(-45)=-44 | | 3 | -15 | 3+(-15)=-12 | | 5 | -9 | 5+(-9)=-4 | | -1 | 45 | -1+45=44 | | -3 | 15 | -3+15=12 | | -5 | 9 | -5+9=4 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.
 Combine like terms. Or factor out the common term 
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Answer:
So  factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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| Solved by pluggable solver: SOLVE quadratic equation with variable |
Quadratic equation  (in our case  ) has the following solutons:
For these solutions to exist, the discriminant  should not be a negative number.
First, we need to compute the discriminant :  .
Discriminant d=196 is greater than zero. That means that there are two solutions:  .
Quadratic expression  can be factored:
Again, the answer is: 1.66666666666667, -3.
Here's your graph:
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Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! 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:
a + b + c = -8
4a + 2b + c = 5
9a + 3b + c = 24
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:
a + b + c = -8
4a + 2b + c = 5
9a + 3b + c = 24
becomes:
3 + 4 + (-15) = -8
4*3 + 2*4 + (-15) = 5
9*3 + 3*4 + (-15) = 24
simplifying, you get:
3 + 4 + (-15) = -8
12 + 8 + (-15) = 5
27 + 12 + (-15) = 24
combining like terms, you get:
-8 = -8
5 = 5
24 = 24
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.
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