f(x)=x^2+ax+b. If f(1)=3 and one of the roots of the equation f(x)=0 doubles the other find the values of a and b
From the given information, we have:
1. f(1) = 3 -> 1^2 + a*1 + b = 3 -> a + b = 2.
We also know that one root of the equation f(x)=0 doubles the other. So let's put those roots in the form k and 2k, where k is the single root.
Then the quadratic can also be written in the form:
f(x) = c * (x - k) * (x - 2k).
Let's apply Vieta's Formulas, which state that the sum of the roots of the quadratic equation is equal to the opposite of the quadratic coefficient, and the product of the roots is equal to the constant of the equation. That is:
- k - 2k = -a
- 3k = -a
k = a/3
and
k * 2k = b
2k^2 = b
Now we can substitute the value of k from the first equation into the second to solve for b:
2 * (a/3)^2 = b
Now we have a system of linear equations:
a + b = 2
b = 2 * (a/3)^2
We can solve this system of equations. First, solve the second equation for a:
a = 3 * sqrt(b/2)
Plug in this expression for a into the first equation:
3 * sqrt(b/2) + b = 2
Square both sides to eliminate the square root:
9 * (b/2) + 6 * sqrt(b/2) * b + b^2 = 4
Combine terms:
(9/2)b + 6 * sqrt(b/2) * b + b^2 = 4
Now, square again to eliminate the remaining square root:
(81/4)b^2 + 54 * (b/2) * b + 12 * b^3 = 16
Combining terms we get:
12*b^3 + 54*b^2 + (81/4)*b = 16
One solution is b = 0, but it's not valid because the double root condition will not be satisfied with both roots equal to zero. We have to solve for the other value of b.
Let's look for a solution of the form b = r^6, for some r. Plugging in b = r^6 in the equation:
12*r^18 + 54*r^12 + (81/4)*r^6 = 16
Now, let x = r^6, so we have:
12x^3 + 54x^2 + (81/4)x = 16
Multiply by 4 to get an integer polynomial:
48x^3 + 216x^2 + 81x = 64
By trial and error or numerical methods, we find x = 1/3 as a root of this equation. Thus, b = r^6 = (1/3)^6 = 1/729.
Now we can find the corresponding value of a using a + b = 2:
a = 2 - b = 2 - 1/729 = 1458/729
So the values of a and b are:
a = 1458/729
b = 1/729