Question

2. Determine the points of intersection of each pair of functions. a) y = 4x2 – 15x + 20 and y = 5x – 4

Answer 1

In order to determine the points of intersection proceed as follow:

Equal both equations:

`4x^2-15x+20=5x-4`

Write the previous equation as an standard quadratic equation:

`\begin{gathered} 4x^2-15x+20=5x-4 \\ 4x^2-15x-5x+20+4=0 \\ 4x^2-20x+24=0 \\ x^2-5x+6=0 \end{gathered}`

to obtain the last equation you divide by 4 both sides.

Now, use the quadratic formula, with a = 1, b = -5 and c = 6, to find the solution for x:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

`\begin{gathered} x=\frac{-(-5)\pm\sqrt[]{(-5)^2-4(1)(6)}}{2(1)} \\ x=\frac{5\pm\sqrt[]{25-24}}{2} \\ x=(5\pm1)/(2) \\ x_1=(5-1)/(2)=(4)/(2)=2 \\ x_2=(5+1)/(2)=(6)/(2)=3 \end{gathered}`

The previous solutions mean that for the values of x = 2 and x = 3 the given functions intersect each other.

By replacing the values of x into any of the functions, for instance, in

y = 5x - 4, you get:

y = 5(2) - 4 = 10 - 4 = 6

y = 5(3) - 4 = 15 - 4 = 11

Then, the points of intersection are:

(2 , 6)

(3 , 11)

The graph is shown below: