Calculate Negative Area of f(x) = x² - 4: Below X-Axis Region

Q: Why does the quadratic have negative values if it contains x²?

Great question! Even though $ x^2 $ is always positive, we're subtracting 4. So when \( x^2 , the result becomes negative. Think of it as: small squares minus 4 give negative results!

Q: How do I know which intervals to test?

The roots divide the number line into sections. For roots at x = -2 and x = 2, you get three intervals: $ (-∞, -2) $, $ (-2, 2) $, and $ (2, ∞) $. Pick any number from each interval as your test point.

Q: What if I pick a different test point?

Any test point within the same interval will give the same sign! For example, in $ (-2, 2) $, you could test x = 0, x = 1, or x = -1 - they'll all give negative results.

Q: Why don't we include the roots x = -2 and x = 2 in our answer?

At the roots, $ f(x) = 0 $, which is neither positive nor negative. Since we want strictly negative values, we use open intervals with

Q: How can I visualize this problem?

Picture a U-shaped parabola that crosses the x-axis at x = -2 and x = 2. The negative area is the part of the curve that dips below the x-axis, which happens between these two crossing points.

Quadratic Inequalities with Root Analysis

Find the negative area of the function

$f(x)=x^2-4$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 First, we find where the function is negative.

00:10 The X squared term is positive, which means the function's graph is smiling or opening upwards.

00:17 The negative domain is below the X axis.

00:22 To find where it crosses the X axis, set Y equals zero.

00:27 Now, let's isolate X.

00:31 Next, extract the square root.

00:35 Remember, when extracting the root, there are two solutions: a positive and a negative.

00:42 These are the points where the function intersects the X axis.

00:48 Let's mark these intersection points on the X axis.

01:01 The positive domain is above the X axis.

01:06 And the negative domain is below the X axis.

01:12 Now, locate where the function is negative.

01:16 And that's how we find the solution to the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the negative area of the function

$f(x)=x^2-4$

Step-by-step solution

To determine the interval where the function $f(x) = x^2 - 4$ is negative, follow these steps:

Step 1: Identify the roots of the function by solving $x^2 - 4 = 0$ .
$x^2 = 4$ yields $x = 2$ and $x = -2$ .
Step 2: Consider the intervals created by these roots: $(-\infty, -2)$ , $(-2, 2)$ , and $(2, \infty)$ .
Step 3: Determine the sign of $f(x)$ in each interval by selecting a test point from each:

For $x = 0$ in $(-2, 2)$ :
$f(0) = 0^2 - 4 = -4$ (Negative)
For $x = -3$ in $(-\infty, -2)$ :
$f(-3) = (-3)^2 - 4 = 9 - 4 = 5$ (Positive)
For $x = 3$ in $(2, \infty)$ :
$f(3) = 3^2 - 4 = 9 - 4 = 5$ (Positive)

Step 4: The function $f(x)$ is negative only in the interval $(-2, 2)$ .

Consequently, the interval where the function has negative values is $-2 < x < 2$ , which aligns with choice 2 in the provided options.

Final Answer

$-2 < x < 2$

Key Points to Remember

Essential concepts to master this topic

Root Finding: Set function equal to zero and solve for x-intercepts
Sign Testing: Check test points in each interval: f(0) = -4 (negative)
Verification: Function is negative between roots where graph dips below x-axis ✓

Common Mistakes

Avoid these frequent errors

Assuming quadratic is always positive or confusing inequality direction
Don't assume $x^2 - 4$ is always positive just because it has $x^2$ = wrong sign analysis! The parabola opens upward but dips below the x-axis between its roots. Always find the roots first, then test points in each interval to determine the actual sign.

Practice Quiz

Test your knowledge with interactive questions

Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

Why does the quadratic have negative values if it contains x²?

Great question! Even though $x^2$ is always positive, we're subtracting 4. So when $x^2 < 4$ , the result becomes negative. Think of it as: small squares minus 4 give negative results!

How do I know which intervals to test?

The roots divide the number line into sections. For roots at x = -2 and x = 2, you get three intervals: $(-∞, -2)$ , $(-2, 2)$ , and $(2, ∞)$ . Pick any number from each interval as your test point.

What if I pick a different test point?

Any test point within the same interval will give the same sign! For example, in $(-2, 2)$ , you could test x = 0, x = 1, or x = -1 - they'll all give negative results.

Why don't we include the roots x = -2 and x = 2 in our answer?

At the roots, $f(x) = 0$ , which is neither positive nor negative. Since we want strictly negative values, we use open intervals with < symbols, not ≤.

How can I visualize this problem?

Picture a U-shaped parabola that crosses the x-axis at x = -2 and x = 2. The negative area is the part of the curve that dips below the x-axis, which happens between these two crossing points.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime