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

Q: Why isn't the answer 'no solution' if the quadratic touches the x-axis at only one point?

Great question! When a quadratic has a double root, it means the parabola just touches the x-axis at one point but doesn't cross it. Since the coefficient of $ x^2 $ is positive (2), the parabola opens upward, so it's positive everywhere except at that touching point.

Q: How do I know the parabola opens upward or downward?

Look at the coefficient of $ x^2 $! If it's positive (like +2 in our problem), the parabola opens upward like a smile. If it's negative, it opens downward like a frown.

Q: What does x ≠ 3 mean exactly?

It means all real numbers except 3. You can write this as $ (-\infty, 3) \cup (3, \infty) $ in interval notation. The inequality is true for every value you can think of, just not when x equals exactly 3.

Q: Do I always need to factor when solving quadratic inequalities?

Not always, but it helps! Factoring shows you exactly where the quadratic equals zero, which are the critical points. You can also use the quadratic formula, but factoring makes the intervals clearer.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Solve the following equation:

$2x^2-12x+18>0$

2

Step-by-step solution

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:

$x ≠ 3$

3

Final Answer

$x ≠ 3$

Key Points to Remember

Essential concepts to master this topic

Perfect Square Rule: When $(x-a)^2 > 0$ , solution excludes only the vertex
Factor Method: $2x^2-12x+18 = 2(x-3)^2$ gives double root at x = 3
Interval Check: Test x = 0: $2(0)^2-12(0)+18 = 18 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Assuming no solution exists when quadratic equals zero at one point
Don't think 'no solution' when you get a perfect square like $(x-3)^2 = 0$ = wrong conclusion! The parabola still opens upward and is positive everywhere except at the vertex. Always remember that $(x-3)^2 > 0$ for all x except x = 3.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why isn't the answer 'no solution' if the quadratic touches the x-axis at only one point?

+

Great question! When a quadratic has a double root, it means the parabola just touches the x-axis at one point but doesn't cross it. Since the coefficient of $x^2$ is positive (2), the parabola opens upward, so it's positive everywhere except at that touching point.

How do I know the parabola opens upward or downward?

+

Look at the coefficient of $x^2$ ! If it's positive (like +2 in our problem), the parabola opens upward like a smile. If it's negative, it opens downward like a frown.

What does x ≠ 3 mean exactly?

+

It means all real numbers except 3. You can write this as $(-\infty, 3) \cup (3, \infty)$ in interval notation. The inequality is true for every value you can think of, just not when x equals exactly 3.

Do I always need to factor when solving quadratic inequalities?

+

Not always, but it helps! Factoring shows you exactly where the quadratic equals zero, which are the critical points. You can also use the quadratic formula, but factoring makes the intervals clearer.

How do I test if my solution is correct?

+

Pick any value except 3 (like x = 0 or x = 5) and substitute into the original inequality. For x = 0: $2(0)^2-12(0)+18 = 18 > 0$ ✓. Then verify x = 3 makes it equal zero, not greater than zero.

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

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