To solve the quadratic inequality [tex]\( 4x^2 > 12x - 9 \)[/tex], follow these steps:
1. Write the inequality in standard form:
The given inequality is:
[tex]\[
4x^2 > 12x - 9
\][/tex]
Subtract [tex]\( 12x - 9 \)[/tex] from both sides to bring everything to one side:
[tex]\[
4x^2 - 12x + 9 > 0
\][/tex]
2. Solve the corresponding equality:
Solve the quadratic equation:
[tex]\[
4x^2 - 12x + 9 = 0
\][/tex]
To find the roots of this quadratic equation, use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 9 \)[/tex]:
[tex]\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(9)}}{2(4)}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{12 \pm \sqrt{144 - 144}}{8}
\][/tex]
Since [tex]\( 144 - 144 = 0 \)[/tex]:
[tex]\[
x = \frac{12 \pm 0}{8} = \frac{12}{8} = 1.5
\][/tex]
Thus, we have one repeated root, [tex]\( x = 1.5 \)[/tex].
3. Analyze the intervals around the root:
The quadratic [tex]\( 4x^2 - 12x + 9 \)[/tex] represents a parabola that opens upwards because the coefficient of [tex]\( x^2 \)[/tex] (which is 4) is positive.
Since the quadratic [tex]\( 4x^2 - 12x + 9 \)[/tex] is always non-negative and equals zero at [tex]\( x = 1.5 \)[/tex], it is positive for [tex]\( x < 1.5 \)[/tex] and [tex]\( x > 1.5 \)[/tex].
4. Write the solution in interval notation:
The inequality [tex]\( 4x^2 - 12x + 9 > 0 \)[/tex] is satisfied for values of [tex]\( x \)[/tex] where the quadratic expression is positive. This happens on the intervals:
[tex]\[
(-\infty, 1.5) \cup (1.5, \infty)
\][/tex]
Thus, the solution to the inequality in interval notation is:
[tex]\[
(-\infty, 1.5) \cup (1.5, \infty)
\][/tex]
