Solve Quadratic Inequality: 2x²-12x+18 Greater Than Zero

To solve this problem, let's first find the solution to the related equation:

• Solve the equation $2x^2 - 12x + 18 = 0$.

The quadratic equation is:

$2x^2 - 12x + 18 = 0$.

Dividing the entire equation by 2 simplifies it to:

$x^2 - 6x + 9 = 0$.

This factors easily as a perfect square:

$(x - 3)^2 = 0$.

This gives a double root at

$x = 3$.

• Analyzing intervals:

The roots of the equation tell us that the parabola touches the x-axis at $x = 3$, and this point is where the inequality would potentially change sign. Since it's a perfect square, the expression $x^2 - 6x + 9 = 0$ means the quadratic doesn't cross the x-axis and is zero at $x = 3$. Therefore, for the inequality $2x^2 - 12x + 18 > 0$, we test the intervals:

Test for $x < 3$ and $x > 3$:

• As the parabola opens upwards (coefficient of $x^2$ is positive), the inequality $2x^2 - 12x + 18 > 0$ holds for all real $x$ except $x = 3$.

Thus, the inequality is true for all $x$ except when $x = 3$.

The correct solution is:

$x ≠ 3$