To solve the inequality $x^2 - 2x + 5 > 0$, we first analyze the quadratic expression.

Step 1: Solve $x^2 - 2x + 5 = 0$ to find its roots

The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

For the equation $x^2 - 2x + 5 = 0$:

• $a = 1$
• $b = -2$
• $c = 5$

$\Delta = (-2)^2 - 4(1)(5) = 4 - 20 = -16$

Since the discriminant is negative ($\Delta = -16$), there are no real roots, meaning the quadratic does not touch the x-axis. The parabola opens upwards because the coefficient of $x^2$ is positive.

Step 2: Analyze the quadratic expression

Since the parabola does not intersect the x-axis and opens upwards, the expression $x^2 - 2x + 5$ is always positive for all real values of $x$.

Conclusion:

The inequality $x^2 - 2x + 5 > 0$ is true for all real values of $x$.

Do you need more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the discriminant help in analyzing quadratic equations?
2. What does the sign of the quadratic coefficient indicate about the parabola's direction?
3. How would you solve a quadratic inequality with real roots?
4. How can you determine the vertex of the quadratic function $x^2 - 2x + 5$?
5. What is the significance of complex roots in quadratic equations?

Tip: When the discriminant is negative, the quadratic expression does not have real roots and will always be positive (if $a > 0$) or always negative (if $a < 0$).