At which of the following points do the two equations f(x)=3x2+5 and g(x)=4x+4 intersect?
I am confused on this one could anyone help me?
thank you! :'0
they intersect when the two functions are equal, right?
3x^2 + 5 = 4x+4
3x^2 - 4x + 1 = 0
(3x-1)(x-1) = 0
To find the intersection points between the two equations f(x) = 3x^2 + 5 and g(x) = 4x + 4, we need to set the two equations equal to each other and solve for x.
Setting 3x^2 + 5 equal to 4x + 4, we have:
3x^2 + 5 = 4x + 4
Rearranging the equation:
3x^2 - 4x + 1 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, the equation can be factored as:
(3x - 1)(x - 1) = 0
Setting each factor equal to zero:
3x - 1 = 0 -----> x = 1/3
x - 1 = 0 -----> x = 1
So, the two equations intersect at the x-values x = 1/3 and x = 1.
To find the corresponding y-values of the intersection points, we can substitute these x-values back into either f(x) or g(x). Let's use f(x).
For x = 1/3:
f(1/3) = 3(1/3)^2 + 5
f(1/3) = 1 + 5
f(1/3) = 6
So, when x = 1/3, the y-coordinate is 6.
For x = 1:
f(1) = 3(1)^2 + 5
f(1) = 3 + 5
f(1) = 8
When x = 1, the y-coordinate is 8.
Therefore, the two equations f(x) = 3x^2 + 5 and g(x) = 4x + 4 intersect at the points (1/3, 6) and (1, 8).
To find the points of intersection between the two equations f(x) = 3x^2 + 5 and g(x) = 4x + 4, we need to set the two equations equal to each other and solve for x.
Step 1: Set the equations equal to each other:
3x^2 + 5 = 4x + 4
Step 2: Rearrange the equation to get it in standard quadratic form:
3x^2 - 4x + 1 = 0
Step 3: Solve the quadratic equation. There are multiple ways to solve quadratic equations, but one common method is factoring. However, in this case, the factors are not easily recognizable, so we can use the quadratic formula.
The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For the equation 3x^2 - 4x + 1 = 0, a = 3, b = -4, and c = 1. Substitute these values into the quadratic formula:
x = (-(-4) ± sqrt((-4)^2 - 4(3)(1))) / (2(3))
Simplifying further:
x = (4 ± sqrt(16 - 12)) / 6
x = (4 ± sqrt(4)) / 6
Step 4: Simplify and find the values of x:
x = (4 ± 2) / 6
This gives us two potential values for x:
x1 = (4 + 2) / 6 = 6 / 6 = 1
x2 = (4 - 2) / 6 = 2 / 6 = 1/3
Step 5: Substitute the values of x back into either of the original equations to find the corresponding y-values. Let's use f(x) = 3x^2 + 5:
For x = 1:
f(1) = 3(1)^2 + 5
f(1) = 3 + 5
f(1) = 8
For x = 1/3:
f(1/3) = 3(1/3)^2 + 5
f(1/3) = 3(1/9) + 5
f(1/3) = 1/3 + 5
f(1/3) = 16/3
Therefore, the two equations f(x) = 3x^2 + 5 and g(x) = 4x + 4 intersect at the points (1, 8) and (1/3, 16/3). These are the solutions to the system of equations.