Question

Solve the polynomial inequality and express in interval notation? `x^2-2x-15<0`

the solution is shown as (-3,5) but why does this problem not use negative infinity or infinity? is there something i'm missing?

Answer 1

A parabola that opens upward can only be less than zero in the interval between the roots.

Explanation:

Please observe that the coefficient of the `x^2` term is greater than 0; this means that the parabola that the equation `y = x^2-2x-15` describes opens upward (as shown in the following graph)

graph{y = x^2-2x-15 [-41.1, 41.1, -20.54, 20.57]}

Please look at the graph and observe that a parabola that opens upward can only be less than zero in the interval between but not including the roots.

The roots of the equation `x^2-2x-15 = 0` can be found by factoring:

`(x +3)(x-5)=0`

`x = -3 and x = 5`

The value of the quadratic is less than zero between these two numbers, `(-3,5)`.

Please look at the graph:

The region in red is the region where the values of y are less than zero; the corresponding x values is the region between the two roots. This is always the case for a parabola of this type. The region in blue contains the y values where the corresponding x values would contain `-oo` but the y values in the region are NEVER less than zero. Similarly, the region in green contains the y values where the corresponding x values would contain `+oo` but the y values in the region are NEVER less than zero.

When you have a parabola that opens upward and the parabola has roots, the region between the two roots is the region that is less than zero; the domain of this region is NEVER bounded by `-oo` or `+oo`.