The solutions to the system of equations involving quadratic function f(x) and linear function g(x) are (-1,5) and (4/3,29/3), respectively.
To find the solution to the given system of equations:
Given quadratic equation: f(x) = 3x^2 + x + 3
The two point formula is: y-y₁ = ((y₂-y₁)/(x₂-x₁))*(x-x₁)
We take the points g(2) = 11, g(1) = 9
g(x) - g(1) = ((g(2)-g(1))/(2-1))*(x-1)
or, g(x) - 9 = ((11-9)/(2-1))*(x-1)
or, g(x) - 9 = 2(x-1)
or, g(x) = 2x - 2 + 9 = 2x + 7
g(x) = 2x +7, is the linear function g(x)
We are asked to solve the system of equations f(x) and g(x).
To get the solution, we must first determine what is the common solution to both f(x) and g(x).
For that, we equate f(x) and g(x).
3x² + x + 3 = 2x + 7
or, 3x² - x - 4 = 0
or, 3x² + 3x - 4x - 4 = 0
or, 3x(x+1) -4(x+1) = 0
or, (3x-4)(x+1) = 0
∴ Either 3x-4=0 ⇒ x = 4/3
or, x+1=0 ⇒ x = -1.
g(-1) = 5 (from the table)
f(-1) = 3(-1)² + (-1) + 3 = 3 - 1 + 3 = 5
g(4/3) = 2(4/3) + 7 = 8/3 + 21/3 = 29/3
f(4/3) = 3*(4/3)² + (4/3) + 3 = 16/3 + 4/3 + 9/3 = 29/3
∴ f(-1) = g(-1) and f(4/3) = g(4/3).
Therefore, the solution to the system of equations that includes quadratic function f(x) and linear function g(x) is (-1,5) and (4/3,29/3).
Know more about polynomial functions here:
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The correct question is given below:
What is a solution to the system of equations that includes quadratic function f(x) and linear function g(x)?
f(x) = 3x^2 + x + 3
