Solve y=x²: Finding Values Where Function is Negative

Q: Why can't $ x^2 $ ever be negative?

When you square any number, you multiply it by itself. A negative times a negative gives a positive, and a positive times a positive stays positive. So $ x^2 \geq 0 $ always!

Q: What about when x is negative like x = -4?

Even when x is negative, $ x^2 $ becomes positive! For example: $ (-4)^2 = (-4) \times (-4) = 16 > 0 $. The square removes the negative sign.

Q: So the answer is 'No x' then?

Exactly! There are no real values of x where $ x^2 . The function \( y = x^2 $ is always greater than or equal to zero.

Q: What's the minimum value of $ x^2 $ ?

The minimum value is 0, which occurs when $ x = 0 $. At this point, $ 0^2 = 0 $, and for any other x value, $ x^2 > 0 $.

Q: How do I remember this for tests?

Think: "Squares are never negative!" Whether you start with positive or negative numbers, squaring always gives you a positive result or zero. It's a fundamental property of real numbers.

Quadratic Functions with Sign Analysis

Given the function:

$y=x^2$

Determine for which values of x $f(x) < 0$ holds

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the function:

$y=x^2$

Determine for which values of x $f(x) < 0$ holds

Step-by-step solution

To determine for which values of $x$ the function $y = x^2$ satisfies $f(x) < 0$ , we need to analyze the nature of the quadratic function.

Step 1: Recognize the function $y = x^2$ is parabolic and opens upwards. For any real number $x$ , $x^2$ is always non-negative, i.e., $x^2 \geq 0$ .

Step 2: Since squaring any real number results in a value greater than or equal to zero, it is not possible for $x^2$ to be less than zero.

Conclusion: Therefore, there are no real values of $x$ for which $x^2 < 0$ . The correct conclusion is that no $x$ satisfies $x^2 < 0$ .

Final Answer

$x\ne0$

Key Points to Remember

Essential concepts to master this topic

Rule: Squaring any real number always gives non-negative result
Technique: Analyze $x^2 \geq 0$ for all real x values
Check: Test specific values: $(-3)^2 = 9 > 0$ and $3^2 = 9 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking negative inputs give negative outputs
Don't assume $x^2 < 0$ when x is negative = wrong conclusion! Squaring removes the negative sign, so $(-5)^2 = 25 > 0$ . Always remember that squaring any real number produces a positive result or zero.

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

FAQ

Everything you need to know about this question

Why can't $x^2$ ever be negative?

When you square any number, you multiply it by itself. A negative times a negative gives a positive, and a positive times a positive stays positive. So $x^2 \geq 0$ always!

What about when x is negative like x = -4?

Even when x is negative, $x^2$ becomes positive! For example: $(-4)^2 = (-4) \times (-4) = 16 > 0$ . The square removes the negative sign.

So the answer is 'No x' then?

Exactly! There are no real values of x where $x^2 < 0$ . The function $y = x^2$ is always greater than or equal to zero.

What's the minimum value of $x^2$ ?

The minimum value is 0, which occurs when $x = 0$ . At this point, $0^2 = 0$ , and for any other x value, $x^2 > 0$ .

How do I remember this for tests?

Think: "Squares are never negative!" Whether you start with positive or negative numbers, squaring always gives you a positive result or zero. It's a fundamental property of real numbers.

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Solve y=x²: Finding Values Where Function is Negative

Quadratic Functions with Sign Analysis

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't $x^2$ ever be negative?

What about when x is negative like x = -4?

So the answer is 'No x' then?

What's the minimum value of $x^2$ ?

How do I remember this for tests?

🌟 Unlock Your Math Potential

More Questions

Positive and Negative Domains

Determine X for the Inequality: f(x) = x² > 0

Identifying Values of X for Which 5x² < 0: A Quadratic Inquiry

Solve y = -0.9x²: Finding Values Where Function is Positive

Solve the Quadratic Function Inequality: When is y = -0.9x² Less Than Zero?

Solve y=x²: Finding Values Where Function is Negative

Quadratic Functions with Sign Analysis

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't x2 x^2 x2 ever be negative?

What about when x is negative like x = -4?

So the answer is 'No x' then?

What's the minimum value of x2 x^2 x2?

How do I remember this for tests?

🌟 Unlock Your Math Potential

More Questions

Positive and Negative Domains

Determine X for the Inequality: f(x) = x² > 0

Identifying Values of X for Which 5x² < 0: A Quadratic Inquiry

Solve y = -0.9x²: Finding Values Where Function is Positive

Solve the Quadratic Function Inequality: When is y = -0.9x² Less Than Zero?

Why can't $x^2$ ever be negative?

What's the minimum value of $x^2$ ?