Determine x-Values for the Quadratic: y = 3/4x² < 0

Q: Why can't this quadratic function ever be negative?

Because any number squared is non-negative! Whether x is positive, negative, or zero, $ x^2 \geq 0 $. Since we're multiplying by the positive coefficient $ \frac{3}{4} $, the result stays non-negative.

Q: What if the coefficient was negative instead?

Great question! If we had $ y = -\frac{3}{4}x^2 $, then the function would be negative for all $ x \neq 0 $ because we'd be multiplying a positive $ x^2 $ by a negative coefficient.

Q: Does this mean the function equals zero somewhere?

Yes! The function equals zero when $ x = 0 $ because $ \frac{3}{4}(0)^2 = 0 $. This is the only point where it's neither positive nor negative.

Q: How do I remember that squared terms are always non-negative?

Think of real examples: $ 3^2 = 9 $ (positive) and $ (-3)^2 = 9 $ (also positive!). The negative sign disappears when you square because negative × negative = positive.

Q: What's the difference between ≥ 0 and > 0?

$ f(x) \geq 0 $ means greater than OR equal to zero$ f(x) > 0 $ means strictly greater than zeroOur function equals zero at $ x = 0 $, so we use $ \geq $

Quadratic Inequalities with Non-negative Values

Given the function:

$y=\frac{3}{4}x^2$

Determine for which values of x $f\left(x\right) < 0$ 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:

$y=\frac{3}{4}x^2$

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

Step-by-step solution

To solve this problem, we'll consider the given quadratic function $y = \frac{3}{4}x^2$ and analyze when it could be negative.

Let's break this down:

The expression $x^2$ represents the square of $x$ , which is always non-negative for any real number $x$ .
Since the coefficient of $x^2$ , which is $\frac{3}{4}$ , is positive, multiplying $x^2$ by this constant results in a non-negative value.

Combining these observations:

Since $x^2 \geq 0$ for all $x$ , it follows that $\frac{3}{4}x^2 \geq 0$ for all $x$ .
Thus, the function $y = \frac{3}{4}x^2$ will always be non-negative, meaning it can never be less than zero.

Therefore, there are no values of $x$ for which $f(x) < 0$ .

Based on this analysis, the correct answer is that there are No $x$ for which the expression is negative.

Final Answer

$x\ne0$

Key Points to Remember

Essential concepts to master this topic

Rule: Squared terms are always non-negative for real numbers
Technique: Since $x^2 \geq 0$ , then $\frac{3}{4}x^2 \geq 0$ always
Check: Test any value: $\frac{3}{4}(-2)^2 = \frac{3}{4}(4) = 3 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking negative x-values make the function negative
Don't assume that negative x-values give negative results = wrong conclusion! When you square a negative number, it becomes positive, so $(-3)^2 = 9$ . Always remember that $x^2 \geq 0$ for ANY real number x.

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

FAQ

Everything you need to know about this question

Why can't this quadratic function ever be negative?

Because any number squared is non-negative! Whether x is positive, negative, or zero, $x^2 \geq 0$ . Since we're multiplying by the positive coefficient $\frac{3}{4}$ , the result stays non-negative.

What if the coefficient was negative instead?

Great question! If we had $y = -\frac{3}{4}x^2$ , then the function would be negative for all $x \neq 0$ because we'd be multiplying a positive $x^2$ by a negative coefficient.

Does this mean the function equals zero somewhere?

Yes! The function equals zero when $x = 0$ because $\frac{3}{4}(0)^2 = 0$ . This is the only point where it's neither positive nor negative.

How do I remember that squared terms are always non-negative?

Think of real examples: $3^2 = 9$ (positive) and $(-3)^2 = 9$ (also positive!). The negative sign disappears when you square because negative × negative = positive.

What's the difference between ≥ 0 and > 0?

$f(x) \geq 0$ means greater than OR equal to zero
$f(x) > 0$ means strictly greater than zero
Our function equals zero at $x = 0$ , so we use $\geq$

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