We are given the inequality $x^2 - 10x + c > 0$. We need to find the range of values for the constant $c$ such that this inequality holds true for all real numbers $x$.

1. Problem Description

We are given the inequality $x^2 - 10x + c > 0$. We need to find the range of values for the constant $c$ such that this inequality holds true for all real numbers $x$.

2. Solution Steps

The inequality $x^2 - 10x + c > 0$ must hold for all real $x$. This means the quadratic equation $x^2 - 10x + c = 0$ must have no real roots. If it had real roots, then there would be values of $x$ for which the quadratic is zero or negative.

A quadratic equation $ax^2 + bx + c = 0$ has no real roots if its discriminant is negative. The discriminant is given by:

$D = b^2 - 4ac$

In our case, the quadratic is $x^2 - 10x + c$. Thus, $a = 1$, $b = -10$, and the constant term is $c$. We want the discriminant to be negative:

$(-10)^2 - 4(1)(c) < 0$

$100 - 4c < 0$

$100 < 4c$

$c > \frac{100}{4}$

$c > 25$

Thus, the range of values for $c$ that makes the inequality $x^2 - 10x + c > 0$ true for all real numbers $x$ is $c > 25$.

3. Final Answer

$c > 25$