How do you find all real values of x such that f(x)=0?
Example 1: f(x)=15-3x
Example 2: f(x)= (12-x^2)/5
Example 3: f(X)= x^3 -x
HUH ?
set to 0
15 - 3 x = 0
3 x = 15
x = 5
12 - x^2 = 5*0 = 0
x^2 = 12
x = +/- sqrt 12 = +/- 2 sqrt 3
x^3 - x = 0
x (x^2-1) = 0
x (x-1)(x+1) = 0
x = 0
x = 1
x = -1
To find all real values of x such that f(x) = 0, we can set each given function equal to zero and solve for x.
Example 1:
f(x) = 15 - 3x
Setting f(x) equal to zero:
0 = 15 - 3x
Rearranging the equation:
3x = 15
Dividing both sides by 3:
x = 5
Therefore, the only real value of x that satisfies f(x) = 0 for this example is x = 5.
Example 2:
f(x) = (12 - x^2) / 5
Setting f(x) equal to zero:
0 = (12 - x^2) / 5
Multiplying both sides by 5:
0 = 12 - x^2
Rearranging the equation:
x^2 = 12
Taking the square root of both sides:
x = ±√12
Simplifying the square root:
x = ±2√3
Therefore, the real values of x that satisfy f(x) = 0 for this example are x = -2√3 and x = 2√3.
Example 3:
f(x) = x^3 - x
Setting f(x) equal to zero:
0 = x^3 - x
Factoring out x from the equation:
0 = x(x^2 - 1)
Using the difference of squares, we can further factor:
0 = x(x - 1)(x + 1)
Therefore, the real values of x that satisfy f(x) = 0 for this example are x = 0, x = 1, and x = -1.
In summary:
Example 1: The real value of x that satisfies f(x) = 0 is x = 5.
Example 2: The real values of x that satisfy f(x) = 0 are x = -2√3 and x = 2√3.
Example 3: The real values of x that satisfy f(x) = 0 are x = 0, x = 1, and x = -1.
To find all real values of x such that f(x) = 0, we need to solve the given equation for x. Here's how you can find the solutions for each example:
Example 1: f(x) = 15 - 3x
To find the real values of x where f(x) = 0, we set f(x) equal to 0 and solve for x:
15 - 3x = 0
To isolate x, subtract 15 from both sides:
-3x = -15
Divide both sides by -3:
x = 5
So the only real value of x that makes f(x) = 0 in this example is x = 5.
Example 2: f(x) = (12 - x^2) / 5
To find the real values of x where f(x) = 0, we set f(x) equal to 0 and solve for x:
(12 - x^2) / 5 = 0
To isolate x^2, multiply both sides by 5:
12 - x^2 = 0
Now, subtract 12 from both sides:
-x^2 = -12
To get rid of the negative sign, multiply both sides by -1:
x^2 = 12
To find the square roots, take the square root of both sides:
x = ± √(12)
Simplifying further:
x = ± √(4 * 3)
x = ± 2√3
So the real values of x that make f(x) = 0 in this example are x = 2√3 and x = -2√3.
Example 3: f(x) = x^3 - x
To find the real values of x where f(x) = 0, we set f(x) equal to 0 and solve for x:
x^3 - x = 0
Factoring out x:
x(x^2 - 1) = 0
We have two cases to consider:
Case 1: x = 0
Case 2: x^2 - 1 = 0
For Case 1, x = 0 is a solution.
For Case 2, we solve the quadratic equation:
x^2 - 1 = 0
(x - 1)(x + 1) = 0
This gives us two additional solutions:
x - 1 = 0 --> x = 1
x + 1 = 0 --> x = -1
So the real values of x that make f(x) = 0 in this example are x = 0, x = 1, and x = -1.