To show that the inequality [tex]\( 4x^2 > 20x - 27 \)[/tex] holds for all [tex]\( x \in \mathbb{R} \)[/tex], we will proceed through a detailed, step-by-step solution.
1. Rewriting the Inequality: First, we'll rearrange the given inequality to standard quadratic form:
[tex]\[
4x^2 - 20x + 27 > 0
\][/tex]
2. Analyzing the Quadratic Equation: Let's consider the related quadratic equation:
[tex]\[
4x^2 - 20x + 27 = 0
\][/tex]
3. Calculating the Discriminant: The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex]. For the equation [tex]\( 4x^2 - 20x + 27 = 0 \)[/tex]:
[tex]\[
a = 4, \quad b = -20, \quad c = 27
\][/tex]
The discriminant is:
[tex]\[
\Delta = (-20)^2 - 4 \cdot 4 \cdot 27 = 400 - 432 = -32
\][/tex]
4. Interpretation of the Discriminant: Since the discriminant [tex]\(\Delta = -32\)[/tex] is less than 0, the quadratic equation [tex]\( 4x^2 - 20x + 27 = 0 \)[/tex] has no real roots and its roots are complex numbers.
5. Roots of the Quadratic Equation: Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex], the roots are:
[tex]\[
x = \frac{20 \pm \sqrt{-32}}{8} = \frac{20 \pm \sqrt{32}i}{8} = \frac{20 \pm 4\sqrt{2}i}{8} = 2.5 \pm 0.7071067811865476i
\][/tex]
Therefore, the roots are:
[tex]\[
x = 2.5 + 0.7071067811865476i \quad \text{and} \quad x = 2.5 - 0.7071067811865476i
\][/tex]
These are complex conjugate roots.
6. Conclusion on the Inequality: Since the quadratic equation [tex]\( 4x^2 - 20x + 27 = 0 \)[/tex] has no real roots (the discriminant is negative, and the roots are complex), the quadratic expression [tex]\( 4x^2 - 20x + 27 \)[/tex] does not change sign. Because the leading coefficient (a = 4) is positive, the quadratic expression [tex]\( 4x^2 - 20x + 27 \)[/tex] is always positive for all [tex]\( x \in \mathbb{R} \)[/tex].
Thus, for all [tex]\( x \in \mathbb{R} \)[/tex],
[tex]\[
4x^2 - 20x + 27 > 0
\][/tex]
This completes the proof that the inequality [tex]\( 4x^2 > 20x - 27 \)[/tex] holds for all real numbers [tex]\( x \)[/tex].
