Determine x Values for y = 2x² + 6 When Positive

Given the function:

$y=2x^2+6$

Determine for which values of x is $f\left(x\right) > 0$ true

To solve this problem, let's analyze the quadratic function $y = 2x^2 + 6$:

• The function $y = 2x^2 + 6$ is a parabola opening upwards because the coefficient of $x^2$ is positive ($a = 2$).
• A parabola that opens upwards has its vertex as the lowest point.
• Since the function is in the form $y = ax^2 + c$ with no $x$ term, there are no real roots (our function has no $b$ term, hence no rotations impacting x-intercepts).
• We evaluate $f(x) > 0$: Since the constant term $c = 6$ is positive, this defines a vertical shift of the parabola entirely above the x-axis.
• This implies $y = 2x^2 + 6$ is always greater than zero for any value of $x$.

Therefore, the solution to the problem is all x.