Question

Consider the functions f(x) = 8x + 16 and g(x) = x^2 + 2x + 5.

At what positive integer value of x does the quadratic function, g(x), begin to exceed the linear function, f(x)?

Answer 1

x =8

Explanation:

To answer this question you must find the point at which
`g(x)\geq f(x)`

So, we have:

`x^2 + 2x + 5 \geq 8x + 16`

`x^2 + 2x -8x + 5 -16\geq0\\\\x^2 -6x -11\geq 0`

To solve the quadratic function we use the quadratic formula

±

`(-b \± √(b^2- 4ac))/(2a)`

Where:

`a = 1\\b =-6\\c = -11`

Then:

`(-(-6) \± √((-6)^2- 4(1)(-11)))/(2(1))\\\\x = 7.47\\x = -1.472`

The line cuts the parabola by 2 points, x = -1.472 and x = 7.47.

You can verify that between x = -1.472 and x = 7.47. the line is greater than the parabola, but from x = 7.47, the parabola is always greater than the graph of the line.

Therefore the point sought is:

x = 7.47≈ 8