I don't get how to do this. I'm so lost please help me.
Determine the equation of the parabola with roots 2+√3 and 2-√3, and is passing through the point (2,5)
this is what i did but its the wrong answer:
y= a[x + (2+√3)] [x+(2-√3)]
5= a[2 + (2+√3)] [2+(2-√3)]
5= a[ 4+√3] [4-√3]
5= 12.99a
a= 0.38
therefore,y = 0.38[x+(2+√3)][x+(2-√3)]
but this is wrong
If you know a root and form the factors you have to subtract the root.
e.g. if x = 3, then the corresponding factor is (x-3)
so you should have had
y = a[x - (2+√3)][x-(2-√3)]
= a(x-2-√3)(x-2+√3)
= a(x^2 - 4x +1)
now sub in the point (2,5)
5 = a(4 - 8 + 1)
a = -5/3
y = (-5/3)[x^2 - 4x + 1)
To determine the equation of the parabola, you are on the right track by using the fact that the roots of the quadratic equation are 2+√3 and 2-√3. This implies that the quadratic equation can be written as:
(x - (2+√3))(x - (2-√3)) = 0
First, let's simplify this expression:
(x - (2+√3))(x - (2-√3)) = 0
(x - 2 - √3)(x - 2 + √3) = 0
(x - 2)^2 - (√3)^2 = 0
(x - 2)^2 - 3 = 0
x^2 - 4x + 4 - 3 = 0
x^2 - 4x + 1 = 0
Now, we have the quadratic equation in the form y = ax^2 + bx + c, and we need to find the value of 'a'. To do this, we can use the fact that the parabola passes through the point (2, 5). Substituting these values into the equation, we get:
5 = a(2)^2 + b(2) + c
5 = 4a + 2b + c
Now, we have two equations:
1) x^2 - 4x + 1 = 0
2) 5 = 4a + 2b + c
We can solve this system of equations to find the values of 'a', 'b', and 'c'.