Final answer:
To find the value of a for f(a) equalling -15 in the function f(x)=x²-3x-19, substitute -15 for f(x) and solve the resulting quadratic equation using the quadratic formula, yielding two solutions for a: 4 and -1.
Step-by-step explanation:
To find the value of a for which f(a) = -15, we substitute -15 for f(x) and then solve the quadratic equation:
f(a) = a² - 3a - 19, so
-15 = a² - 3a - 19
We then bring all terms to one side of the equation to get:
a² - 3a - 4 = 0
Now, we can use the quadratic formula, which states that if ax² + bx + c = 0, then x can be solved using x = (-b ± √(b²-4ac)) / (2a).
Here, a = 1, b = -3, and c = -4. Plugging these values into the quadratic formula, we get:
a = (-(-3) ± √((-3)²-4*1*(-4))) / (2*1)
a = (3 ± √(9 + 16)) / 2
a = (3 ± √25) / 2
a = (3 ± 5) / 2
Thus, a has two possible values:
a = (3 + 5) / 2 = 4 or a = (3 - 5) / 2 = -1