Solve for Positive Values: y = 2x² - 4x + 5

Look at the following function:

$y=2x^2-4x+5$

Determine for which values of $x$ the following is is true:

$f\left(x\right)>0$

To determine for which values of $x$ the function $y = 2x^2 - 4x + 5$ is positive, we will analyze its characteristics.

Step 1: Determine the direction of the parabola.
The given quadratic function $y = 2x^2 - 4x + 5$ has a leading coefficient $a = 2$, which is positive. Therefore, the parabola opens upwards.

Step 2: Check for real roots.
To identify where the function might be zero, calculate the discriminant $\Delta = b^2 - 4ac$.
Here, $a = 2$, $b = -4$, $c = 5$.
The discriminant $\Delta = (-4)^2 - 4 \cdot 2 \cdot 5 = 16 - 40 = -24$.
Since the discriminant is negative, the quadratic has no real roots, meaning it doesn't intersect the x-axis.

Step 3: Analyze positivity over the entire domain.
Since the parabola opens upwards and has no real roots, the function does not touch or cross the x-axis. Therefore, $y = 2x^2 - 4x + 5$ is always positive.

Conclusion.
The function $y = 2x^2 - 4x + 5$ is positive for all values of $x$.

Therefore, the solution to the problem is The function is positive for all values of $x$.