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

2. Solution Steps

For the quadratic expression $x^2 - 10x + c$ to be greater than 0 for all real values of $x$, the parabola represented by the quadratic function $f(x) = x^2 - 10x + c$ must open upwards (which it does since the coefficient of $x^2$ is positive) and must not intersect the x-axis. This means the quadratic equation $x^2 - 10x + c = 0$ must have no real roots.

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

$D = b^2 - 4ac$

In our case, the quadratic equation is $x^2 - 10x + c = 0$, so $a = 1$, $b = -10$, and the constant term is $c$. Thus, the discriminant is:

$D = (-10)^2 - 4(1)(c) = 100 - 4c$

For the quadratic equation to have no real roots, the discriminant must be less than 0:

$D < 0$

$100 - 4c < 0$

$100 < 4c$

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

$c > 25$

Therefore, 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

$c > 25$