Question

How you can tell that the quadratic function `y=x^2 +6x+15` has no real zeroes without graphing the function?

Answer 1

When given a quadratic equation in standard form:

`y = ax^2 + bx + c` or `x = ay^2 + by + c`

Check the value of the determinant:

`b^2 - 4(a)(c)`

If it is greater than zero, the equation will have two real roots.
If it is equal to zero, the equation will have one root (a.k.a. a repeated root).
If it is less than zero, the equation will have complex conjugate pair of roots.

In the case of the given equation

`b^2 - 4(a)(c) = 6^2 - 4(1)(15) = -24`

This equation will have a complex conjugate pair of roots `(-3 + sqrt6i) and (-3 - sqrt6i)` .