I've been staring at this problem for an hour and have NO IDEA WHAT TO DO!
"A quadratic function in the form f(x)=ax^2+bx+c contains the points (-1,0), (0,2), and (2,0)."
a) determine the values of a,b, and, c
b) Factorise the expression ax^2+bx+c obtained from the function in part (a)
please help! i've tried EVERYTHING :(
Let
f(x)=ax^2+bx+c
from (-1,0), we conclude that
f(-1)=0, or
a(-1)^2 + b(-1) + c = 0......(1)
Similarly, from (0,2), we have
a(0)^2 + b(0) + c = 2........(2)
and from (2,0), we get
a(2)^2 + b(2) + c = 0........(3)
By solving the system of linear equations (1),(2) and (3), you will find the values of a,b and c.
After substituting the numerical values of a,b and c into f(x), you can factorize as required in question (c).
Note: the system of linear equations is easy to solve because if you examine equation (2), it reduces to c=2.
By substituting the value of c in (1) and (3), you only have to solve a system of 2 equations with two unknowns (a and b).
A quadratic function in the form f(x)=ax^2+bx+c contains the points (-1,0), (0,2), and (2,0)."
Another way
y = (x+1)(x-2)(k)
2 = (1)(-2)(k)
2 = -2k
k = -1
so
y = (x+1)(x-2)(-1) (factored already)
so
y = -x^2 + x + 2
y =
It's okay, I'm here to help you! Let's break down the problem step by step.
a) To determine the values of a, b, and c, we will use the given information about the points (-1,0), (0,2), and (2,0). Since these points lie on the quadratic function f(x), we can substitute the x and y values of each point into the equation and solve for the unknowns.
Let's start with the point (-1,0):
f(-1) = a(-1)^2 + b(-1) + c
0 = a - b + c
Next, let's move to the point (0,2):
f(0) = a(0)^2 + b(0) + c
2 = c
Finally, we'll work with the point (2,0):
f(2) = a(2)^2 + b(2) + c
0 = 4a + 2b + c
Now, we have a system of three equations:
0 = a - b + c (equation 1)
2 = c (equation 2)
0 = 4a + 2b + c (equation 3)
We can substitute equation 2 into equations 1 and 3 to solve for a and b.
Substitute equation 2 into equation 1:
0 = a - b + 2 (equation 4)
Substitute equation 2 into equation 3:
0 = 4a + 2b + 2 (equation 5)
Now we have two equations (4 and 5) with two variables (a and b) that we can solve simultaneously.
From equation 4, we can rewrite it as: a = b - 2
Substitute this value of a into equation 5:
0 = 4(b - 2) + 2b + 2
Expand and simplify equation 5:
0 = 4b - 8 + 2b + 2
0 = 6b - 6
Solve for b:
6b = 6
b = 1
Now substitute this value of b into equation 4 to solve for a:
a = (1) - 2
a = -1
Finally, we already found from equation 2 that c = 2.
Therefore, the values of a, b, and c are:
a = -1
b = 1
c = 2
b) To factorize the expression ax^2+bx+c, substitute the values we found into the equation:
-1x^2 + 1x + 2
To factorize this quadratic expression, we need to find two binomials that satisfy the equation. We will use the fact that the product of the first terms must be -1x^2, and the product of the last terms must be +2.
The factors could be:
(-1x + 2) and (1x + 1)
So the factorized expression is:
(-1x + 2)(1x + 1)
I hope this step-by-step explanation helps you understand how to solve the problem. If you have any further questions, feel free to ask!