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?
Hint: This must be a perfect square -- in the form (a + b)^2 or (a - b)^2.
Consider both the cases.
45
solutions are
a-15=30 i.e a=45
a-15=-30 i.e a=-15
therefore, answer is 30
To find the sum of all the possible values of "a" which would give the quadratic equation 25x^2 + (a-15)x + 9 exactly one distinct real root, we need to use the discriminant.
The discriminant (denoted by Δ) is a mathematical term that can be used to determine the nature of the roots of a quadratic equation. For a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are coefficients, the discriminant is given by the formula Δ = b^2 - 4ac.
In this case, a = 25, b = (a-15), and c = 9. Therefore, the discriminant of the quadratic equation is Δ = (a-15)^2 - 4 * 25 * 9.
For the equation to have exactly one distinct real root, the discriminant must be equal to zero, since a discriminant of zero implies that the equation has one root. So, we set Δ = 0 and solve for "a".
(a-15)^2 - 4 * 25 * 9 = 0
Expanding and simplifying:
a^2 - 30a + 225 - 900 = 0
a^2 - 30a - 675 = 0
Now, we can use the quadratic formula to solve for "a":
a = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -30, and c = -675. Plugging in these values:
a = (-(-30) ± √((-30)^2 - 4 * 1 * (-675))) / (2 * 1)
a = (30 ± √(900 + 2700)) / 2
a = (30 ± √3600) / 2
Simplifying further:
a = (30 ± 60) / 2
Therefore, the two possible values of "a" are (30 + 60)/2 = 45 and (30 - 60)/2 = -15.
The sum of all possible values of "a" is 45 + (-15) = 30.
Hence, the sum of all the possible values of "a" for which 25x^2 + (a-15)x + 9 has exactly one distinct real root is 30.