Answer:
To determine if we can use factoring or the Quadratic formula to find the roots of the function F(x) = x^2 + 2x - 8, we need to check if the function is factorable or if it fits the criteria for the Quadratic formula.
Let's start by checking if the function can be factored. For a quadratic function to be factorable, the coefficient of the x^2 term should be 1, and the constant term should be a product of two numbers that add up to the coefficient of the x term.
In our function, the coefficient of the x^2 term is 1, which is good. However, the constant term, -8, cannot be factored into two numbers that add up to 2, the coefficient of the x term. Therefore, we cannot use factoring to find the roots of this function.
Now, let's move on to the Quadratic formula. The Quadratic formula is a useful tool for finding the roots of any quadratic function of the form ax^2 + bx + c = 0, where a, b, and c are constants.
The Quadratic formula is given by: x = (-b ± √(b^2 - 4ac))/(2a)
For our function F(x) = x^2 + 2x - 8, the coefficients are:
a = 1 (coefficient of x^2 term)
b = 2 (coefficient of x term)
c = -8 (constant term)
Now, we can plug these values into the Quadratic formula and solve for x. Let's do that:
x = (-2 ± √(2^2 - 4(1)(-8)))/(2(1))
Simplifying further:
x = (-2 ± √(4 + 32))/2
x = (-2 ± √36)/2
x = (-2 ± 6)/2
Now, we can simplify the equation further by separating it into two solutions:
x1 = (-2 + 6)/2 = 4/2 = 2
x2 = (-2 - 6)/2 = -8/2 = -4
Therefore, the roots of the function F(x) = x^2 + 2x - 8 are x = 2 and x = -4.