Solve y=-7x² > 0: Finding Values Where Function is Positive

Quadratic Inequalities with Negative Coefficients

Given the function y=7x2 y=-7x^2

Determine for which values of x the following holds:

f(x)>0 f\left(x\right) > 0

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Given the function y=7x2 y=-7x^2

Determine for which values of x the following holds:

f(x)>0 f\left(x\right) > 0

2

Step-by-step solution

To solve this problem, let's work through the following steps:

Step 1: Analyze the inequality 7x2>0 -7x^2 > 0 .
Since x20 x^2 \geq 0 for all real x x and x2=0 x^2 = 0 when x=0 x = 0 , we know x2>0 x^2 > 0 when x0 x \neq 0 . However, the expression 7x2 -7x^2 is always less than or equal to zero because multiplying a non-negative x2 x^2 by 7-7 gives a non-positive result.
Therefore, 7x2>0 -7x^2 > 0 cannot be true for any real x x .

Step 2: Conclude based on this analysis.
The only scenario where 7x2 -7x^2 could have been greater than zero is if we had a positive term offsetting it, which is not the case.
Hence, there are no values of x x for which f(x)>0 f(x) > 0 .

The correct answer based on the provided choices is No x.

3

Final Answer

x0 x\ne0

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative coefficient makes parabola open downward, always non-positive
  • Technique: Since 7x20 -7x^2 \leq 0 for all x, no solution exists
  • Check: Test x = 1: 7(1)2=7<0 -7(1)^2 = -7 < 0

Common Mistakes

Avoid these frequent errors
  • Ignoring the negative sign in front of x²
    Don't treat 7x2 -7x^2 like 7x2 7x^2 = positive values! The negative coefficient makes the entire expression non-positive for all real x. Always consider how the negative sign affects the range of the function.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the \( x \)-axis.

The parabola's vertex is marked A.

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

AAAX

FAQ

Everything you need to know about this question

Why can't 7x2 -7x^2 ever be positive?

+

Because x20 x^2 \geq 0 for all real numbers, and multiplying by -7 (negative) makes the result non-positive. The maximum value is 0 when x = 0.

What if the question asked for f(x)0 f(x) \geq 0 instead?

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Then x = 0 would be the only solution! The function equals zero at x = 0 and is negative everywhere else.

How do I know when a quadratic has no positive values?

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Look at the coefficient of x2 x^2 . If it's negative and there's no positive constant term, the parabola opens downward with maximum value ≤ 0.

Could I graph this to see the answer?

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Absolutely! Graph y=7x2 y = -7x^2 and you'll see it's a downward-opening parabola with vertex at (0,0). The entire graph lies on or below the x-axis.

Why is the answer 'No x' and not 'x ≠ 0'?

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Because the question asks when f(x)>0 f(x) > 0 (strictly greater than zero). Since 7x20 -7x^2 \leq 0 for all x, there are no values where it's positive.

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