Answer: x = 3, and x = -4
Explanation:
The roots of a quadratic equation can be found by setting the equation equal to zero and solving for x. In this case, we have the equation f(x) = 2x^2 + 2x - 24.
To find the roots, we need to set f(x) equal to zero:
2x^2 + 2x - 24 = 0.
Now, we can solve this equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the roots are given by:
x = (-b ± √(b^2 - 4ac)) / (2a).
In our equation, a = 2, b = 2, and c = -24. Plugging these values into the quadratic formula, we get:
x = (-2 ± √(2^2 - 4(2)(-24))) / (2(2)).
Simplifying the expression inside the square root:
x = (-2 ± √(4 + 192)) / 4,
x = (-2 ± √196) / 4,
x = (-2 ± 14) / 4.
Now we have two possible values for x:
x1 = (-2 + 14) / 4 = 12/4 = 3,
x2 = (-2 - 14) / 4 = -16/4 = -4.
Therefore, the potential roots of f(x) = 2x^2 + 2x - 24 are x = 3 and x = -4.