0.4x² Inequality Solution: When is the Parabola Above Zero?

Quadratic Inequalities with Vertex Analysis

Given the function:

y=0.4x2 y=0.4x^2

Determine for which values of x f(x)>0 f(x) > 0 holds

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Step-by-step written solution

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1

Understand the problem

Given the function:

y=0.4x2 y=0.4x^2

Determine for which values of x f(x)>0 f(x) > 0 holds

2

Step-by-step solution

To solve this problem, let's break down the given quadratic function:

  • Step 1: The function is y=0.4x2 y = 0.4x^2 . This is a basic quadratic function where a=0.4 a = 0.4 , b=0 b = 0 , and c=0 c = 0 .
  • Step 2: Since a=0.4>0 a = 0.4 > 0 , the parabola opens upwards. The vertex of this parabola is at the point (0,0) (0, 0) .
  • Step 3: Address the condition f(x)>0 f(x) > 0 . The function value y y is zero exactly at the vertex, x=0 x = 0 . For any other real number value of x x , the term x2 x^2 is positive, and therefore 0.4x2 0.4x^2 is also positive.

Since y=0.4x2 y = 0.4x^2 will always be greater than zero for every x≠0 x \neq 0 , the correct set of values for x x where f(x)>0 f(x) > 0 is all x x except x=0 x = 0 . Thus, the solution is expressed as:

x≠0 x \ne 0

3

Final Answer

x≠0 x\ne0

Key Points to Remember

Essential concepts to master this topic
  • Parabola Direction: Coefficient 0.4 > 0 means parabola opens upward
  • Vertex Location: Find where y=0.4x2=0 y = 0.4x^2 = 0 gives vertex (0,0)
  • Verification: Test x = 1: 0.4(1)2=0.4>0 0.4(1)^2 = 0.4 > 0 ✓

Common Mistakes

Avoid these frequent errors
  • Thinking the function is always positive for all x values
    Don't assume 0.4x2>0 0.4x^2 > 0 for ALL x = wrong answer 'All x'! At x = 0, the function equals zero, not greater than zero. Always check the vertex point where the parabola touches the x-axis.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of \( x \) where \( f\left(x\right) > 0 \).

AAABBBCCCX

FAQ

Everything you need to know about this question

Why isn't x = 0 included in the solution?

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At x = 0, we get y=0.4(0)2=0 y = 0.4(0)^2 = 0 . Since we need f(x) > 0 (strictly greater than), zero doesn't count. We need values where the function is positive, not zero.

How do I know the parabola opens upward?

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Look at the coefficient of x2 x^2 ! Since 0.4 > 0, the parabola opens upward like a U-shape. If it were negative, it would open downward like an upside-down U.

What does x ≠ 0 mean exactly?

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x≠0 x \ne 0 means all real numbers except zero. This includes negative numbers like -5, positive numbers like 3.7, and even fractions like 1/2 - just not zero itself.

How can I verify this graphically?

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Graph y=0.4x2 y = 0.4x^2 and look where the curve is above the x-axis. You'll see it touches the x-axis only at (0,0) but is positive everywhere else!

Would the answer change if the coefficient was negative?

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Yes! If we had y=−0.4x2 y = -0.4x^2 , the parabola would open downward. Then f(x) > 0 would have no solutions since the function would always be ≤ 0.

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