Find the zeros of the function algebraically. (Enter your answers as a comma-separated list.)
f(x) = 2x2 − 3x − 9
2x^2 − 3x − 9 = 0
factoring ... (2x + 3)(x - 3) = 0
2x + 3 = 0 ... x = -3/2
x - 3 = 0 ... x = 3
To find the zeros of the function algebraically, we need to solve the equation f(x) = 0.
The given function is f(x) = 2x^2 - 3x - 9.
Setting this equation to zero, we have:
2x^2 - 3x - 9 = 0
To solve this quadratic equation, we can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -3, and c = -9.
Plugging these values into the quadratic formula, we get:
x = (-(-3) ± √((-3)^2 - 4(2)(-9))) / (2(2))
x = (3 ± √(9 + 72)) / 4
x = (3 ± √81) / 4
x = (3 ± 9) / 4
Now we have two possible values for x:
x1 = (3 + 9) / 4 = 12 / 4 = 3
x2 = (3 - 9) / 4 = -6 / 4 = -3/2
Therefore, the zeros of the function algebraically are x = 3 and x = -3/2.
In comma-separated form, the zeros are: 3, -3/2.
To find the zeros of a function algebraically, we need to solve the equation f(x) = 0. In this case, the function is f(x) = 2x^2 - 3x - 9.
To solve the equation, we can either use factoring or the quadratic formula.
1. Factoring: We want to factorize the quadratic equation 2x^2 - 3x - 9 = 0. Unfortunately, this quadratic equation cannot be easily factored into linear binomials. Therefore, we will resort to the quadratic formula.
2. Quadratic Formula: The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Now let's apply the quadratic formula to our equation 2x^2 - 3x - 9 = 0:
a = 2, b = -3, c = -9
x = (-(-3) ± √((-3)^2 - 4(2)(-9))) / (2(2))
= (3 ± √(9 + 72)) / 4
= (3 ± √81) / 4
= (3 ± 9) / 4
Simplifying further, we have two possible solutions:
For x = (3 + 9) / 4:
x = 12 / 4
x = 3
For x = (3 - 9) / 4:
x = -6 / 4
x = -3/2
Therefore, the zeros (or x-intercepts) of the function f(x) = 2x^2 - 3x - 9 are 3 and -3/2.
You can verify this by substituting these values back into the original equation and checking if they make the equation equal to 0.