If the quadratic equation x^2 + bx +c=0 has exactly one solution r, then find b/c.
x^2 + bx + c = (x-r)^2
2r = -b
r^2 = c
b/c = r^2/(-r/2) = -2r
To find the value of b/c, we need to first determine the condition for a quadratic equation to have exactly one solution.
In a quadratic equation of the form ax^2 + bx + c = 0, the discriminant (denoted as Δ) is used to determine the number of solutions:
Δ = b^2 - 4ac
If the discriminant is equal to zero (Δ = 0), then the quadratic equation has exactly one solution.
For the given quadratic equation x^2 + bx + c = 0, we can calculate the discriminant as follows:
Δ = b^2 - 4ac
Since we know that the equation has exactly one solution (r), we have Δ = 0:
0 = b^2 - 4ac
Now, we can substitute x^2 + bx + c = 0 into the equation to find b/c.
0 = b^2 - 4ac = r^2 + br + c
Since we are looking for the ratio b/c, we divide the equation by c:
0 = (b^2/c) - 4a
Therefore, the value of b/c is equal to 4a divided by c.