To solve the quadratic inequality [tex]\(2x^2 - 9x + 4 > 0\)[/tex], we need to follow these steps:
### Step 1: Find the roots of the corresponding quadratic equation
First, solve the equation [tex]\(2x^2 - 9x + 4 = 0\)[/tex]. We can use the quadratic formula to find the roots:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 2\)[/tex], [tex]\(b = -9\)[/tex], and [tex]\(c = 4\)[/tex].
Plugging in these values, we get:
[tex]\[
x = \frac{9 \pm \sqrt{(-9)^2 - 4(2)(4)}}{2(2)}
\][/tex]
[tex]\[
x = \frac{9 \pm \sqrt{81 - 32}}{4}
\][/tex]
[tex]\[
x = \frac{9 \pm \sqrt{49}}{4}
\][/tex]
[tex]\[
x = \frac{9 \pm 7}{4}
\][/tex]
This gives us two roots:
[tex]\[
x = \frac{9 + 7}{4} = \frac{16}{4} = 4
\][/tex]
[tex]\[
x = \frac{9 - 7}{4} = \frac{2}{4} = \frac{1}{2}
\][/tex]
So, the roots of the quadratic equation are [tex]\(x = 4\)[/tex] and [tex]\(x = \frac{1}{2}\)[/tex].
### Step 2: Determine the intervals around the roots
Since the quadratic expression [tex]\(2x^2 - 9x + 4\)[/tex] is a parabola opening upwards (the coefficient of [tex]\(x^2\)[/tex] is positive), the quadratic inequality [tex]\(2x^2 - 9x + 4 > 0\)[/tex] holds true outside the interval defined by the roots.
Thus, the intervals we need to consider are:
1. [tex]\(x < \frac{1}{2}\)[/tex]
2. [tex]\(x > 4\)[/tex]
### Conclusion
The solution to the inequality [tex]\(2x^2 - 9x + 4 > 0\)[/tex] is:
[tex]\[
x \in (-\infty, \frac{1}{2}) \cup (4, \infty)
\][/tex]
Therefore, [tex]\(2x^2 - 9x + 4\)[/tex] is greater than [tex]\(0\)[/tex] for [tex]\(x\)[/tex] in the intervals [tex]\((- \infty, \frac{1}{2})\)[/tex] and [tex]\((4, \infty)\)[/tex].
