To solve the quadratic inequality [tex]\( x^2 - 2x + 37 > 0 \)[/tex], we need to analyze the quadratic expression [tex]\( x^2 - 2x + 37 \)[/tex] and determine over which intervals this inequality holds true. Here's a step-by-step process to achieve that:
1. Identify the quadratic function:
[tex]\[
f(x) = x^2 - 2x + 37
\][/tex]
2. Find the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
[tex]\[
a = 1, \quad b = -2, \quad c = 37
\][/tex]
[tex]\[
\Delta = (-2)^2 - 4 \cdot 1 \cdot 37 = 4 - 148 = -144
\][/tex]
3. Analyze the discriminant:
The discriminant [tex]\( \Delta = -144 \)[/tex] is less than zero. This indicates that the quadratic equation has no real roots.
4. Behavior of the quadratic function:
Since [tex]\( a = 1 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex]) is positive, the parabola opens upwards.
5. Determine the nature of the quadratic function:
Given the upward-opening parabola with no real roots, the quadratic function [tex]\( x^2 - 2x + 37 \)[/tex] is always positive for all [tex]\( x \)[/tex]. The graph of this quadratic will lie entirely above the x-axis.
6. Conclusion - Solve the inequality:
[tex]\[
x^2 - 2x + 37 > 0
\][/tex]
Since the function is always positive for all real [tex]\( x \)[/tex], the solution to the inequality is:
[tex]\[
(-\infty, \infty)
\][/tex]
This means that for any real number [tex]\( x \)[/tex], the inequality [tex]\( x^2 - 2x + 37 > 0 \)[/tex] holds true.
Therefore, the solution in interval notation is:
[tex]\[
(-\infty, \infty)
\][/tex]
