Solve the Quadratic Inequality: 2x² - 12x + 18 < 0

Q: What does it mean when the discriminant equals zero?

When $ \Delta = 0 $, the quadratic has exactly one real root (a repeated root). The parabola touches the x-axis at just one point - the vertex - but doesn't cross it.

Q: Why does the parabola never go below the x-axis here?

Since a = 2 > 0, the parabola opens upward. With $ \Delta = 0 $, it only touches the x-axis at the vertex. An upward-opening parabola that touches but doesn't cross the x-axis stays at or above zero.

Q: How can I visualize this problem?

Picture a U-shaped curve that just barely touches the x-axis at one point. Since it opens upward and never dips below the x-axis, there are no x-values where the function is negative.

Q: What if the question asked for ≤ 0 instead of < 0?

Then there would be a solution! The parabola equals zero at $ x = 3 $ (the vertex). So $ 2x^2 - 12x + 18 \leq 0 $ has solution x = 3.

Q: Is there a pattern for when quadratic inequalities have no solutions?

Yes! For $ ax^2 + bx + c : if a > 0 (opens up) and \( \Delta \leq 0 $, there are no solutions because the parabola never goes below the x-axis.

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 inequality $2x^2 - 12x + 18 < 0$ , we perform the following steps:

Step 1: Compute the discriminant. Here, $a = 2$ , $b = -12$ , and $c = 18$ . Plug these into the discriminant formula:

$\Delta = b^2 - 4ac = (-12)^2 - 4 \times 2 \times 18 = 144 - 144 = 0$

Step 2: Interpret $\Delta = 0$ . A discriminant of zero means the quadratic equation has one repeated real root. The graph of the quadratic is a parabola that touches the x-axis at a single point, at the vertex.
Step 3: Since the parabola described by the quadratic has no interval crossing the x-axis (it only touches it at a point), the inequality $2x^2 - 12x + 18 < 0$ has no solutions. A parabola touching the x-axis at a single vertex point does not lie below the x-axis.

Therefore, there are no values of $x$ for which the quadratic expression $2x^2 - 12x + 18 < 0$ . The function does not attain negative values.

The correct answer to the multiple-choice question is therefore:
The function is negative for all values of x.

3

Final Answer

The function is negative for all values of x.

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $\Delta = 0$ , parabola touches x-axis at one point
Analysis Technique: $\Delta = (-12)^2 - 4(2)(18) = 0$ means single vertex contact
Verification: Check parabola opens upward (a > 0) and never goes below x-axis ✓

Common Mistakes

Avoid these frequent errors

Confusing discriminant zero with having solutions
Don't assume $\Delta = 0$ means the inequality has solutions = wrong conclusion! Zero discriminant means the parabola only touches the x-axis, never crosses below it. Always remember that for f(x) < 0, you need the parabola to actually go below the x-axis.

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

What does it mean when the discriminant equals zero?

+

When $\Delta = 0$ , the quadratic has exactly one real root (a repeated root). The parabola touches the x-axis at just one point - the vertex - but doesn't cross it.

Why does the parabola never go below the x-axis here?

+

Since a = 2 > 0, the parabola opens upward. With $\Delta = 0$ , it only touches the x-axis at the vertex. An upward-opening parabola that touches but doesn't cross the x-axis stays at or above zero.

How can I visualize this problem?

+

Picture a U-shaped curve that just barely touches the x-axis at one point. Since it opens upward and never dips below the x-axis, there are no x-values where the function is negative.

What if the question asked for ≤ 0 instead of < 0?

+

Then there would be a solution! The parabola equals zero at $x = 3$ (the vertex). So $2x^2 - 12x + 18 \leq 0$ has solution x = 3.

Is there a pattern for when quadratic inequalities have no solutions?

+

Yes! For $ax^2 + bx + c < 0$ : if a > 0 (opens up) and $\Delta \leq 0$ , there are no solutions because the parabola never goes below the x-axis.

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Solve the Quadratic Inequality: 2x² - 12x + 18 < 0

Quadratic Inequalities with Zero Discriminant

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