What is the sum of all the possible values of a for which 25x^2 + (a-15)x + 9 has exactly one distinct real root?
To find the sum of all the possible values of 'a' for which the quadratic equation 25x^2 + (a-15)x + 9 has exactly one distinct real root, we can use the discriminant of the quadratic equation.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
For one distinct real root, the discriminant must be equal to zero (D = 0).
Let's substitute the values from our equation into the discriminant formula:
D = (a-15)^2 - 4(25)(9)
Simplifying further,
D = a^2 - 30a + 225 - 900
D = a^2 - 30a - 675
For the equation to have exactly one distinct real root, the discriminant should be zero:
a^2 - 30a - 675 = 0
Now, we need to find the values of 'a' that satisfy this equation. We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Factoring or solving the quadratic equation gives us:
(a - 45)(a + 15) = 0
Setting each factor equal to zero, we get:
a - 45 = 0 or a + 15 = 0
Solving for 'a', we find:
a = 45 or a = -15
Therefore, the possible values of 'a' are 45 and -15.
To find the sum of these possible values, we calculate:
45 + (-15) = 30
Hence, the sum of all possible values of 'a' for which the given quadratic equation has exactly one distinct real root is 30.