Question

first calculate how many points of intersection the following system of equations has f(x)=6x^2 - 4x - 25 and g(x) = 3x - 5second calculate the points of intersection of f(x)= 6x^2-4x-25 and g(x) = 3x - 5

Answer 1

Given the system of equations:

`\begin{gathered} f\mleft(x\mright)=6x^2-4x-25 \\ g\mleft(x\mright)=3x-5 \end{gathered}`

To solve the system of equations, we will solve the following equation:

`f(x)=g(x)`

so,

`6x^2-4x-25=3x-5`

Solve the quadratic equation as follows:

`\begin{gathered} 6x^2-4x-25-3x+5=0 \\ 6x^2-7x-20=0 \\ \end{gathered}`

The last equation will be solved using the general formula:

a = 6, b = -7, c = -20

`\begin{gathered} x=\frac{7\pm\sqrt[]{(-7)^2-4\cdot6\cdot(-20)}}{2\cdot6}=(7\pm23)/(12) \\ \\ x=(7+23)/(12)=2.5 \\ or \\ x=(7-23)/(12)=-(4)/(3) \end{gathered}`

So, there are two points of intersection for the given system

For each value of (x) substitute into g(x) to find the value of (y)

`\begin{gathered} x=2.5\rightarrow y=3\cdot2.5-5=2.5 \\ x=-(4)/(3)\rightarrow y=3\cdot-(4)/(3)-5=-9 \end{gathered}`

So, the answer will be, the point of intersections are:

`(x,y)=\mleft\lbrace(2.5,2.5\mright);(-(4)/(3),-9)\}`