Solve y = x²-4: Finding Values Where Function is Negative

Q: Why do I need to find where the function equals zero first?

The zeros (where $ x^2 - 4 = 0 $) are the boundary points where the function changes from positive to negative. These critical points at $ x = ±2 $ divide the number line into intervals to test.

Q: How do I know which intervals to test?

After finding zeros at $ x = -2 $ and $ x = 2 $, test one point in each interval: before -2, between -2 and 2, and after 2. Pick easy numbers like -3, 0, and 3.

Q: What's the difference between < and ≤ in the answer?

For $ f(x) , use open intervals like \( -2 (excludes endpoints). For \( f(x) ≤ 0 $, use closed intervals like $ -2 ≤ x ≤ 2 $ (includes endpoints).

Q: Can I just look at the graph instead of doing algebra?

Yes! The graph shows where the parabola is below the x-axis (negative values). But always verify algebraically by finding the exact zeros and testing intervals for complete accuracy.

Q: Why is the answer an interval instead of specific values?

Unlike equations that have specific solutions, inequalities have ranges of solutions. Every x-value between -2 and 2 makes the function negative, so the solution is the entire interval.

Q: What if I get a different interval when testing?

Double-check your zeros by solving $ x^2 - 4 = 0 $ correctly. Then carefully substitute your test points: $ f(0) = -4 $ (negative) confirms the middle interval is part of the solution.

Quadratic Inequalities with Sign Analysis

Look at the function below:

$y=x^2-4$

Then determine for which values of $x$ the following is true:

$f\left(x\right) < 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the function below:

$y=x^2-4$

Then determine for which values of $x$ the following is true:

$f\left(x\right) < 0$

Step-by-step solution

To determine for which values of $x$ the function $f(x) = x^2 - 4$ is less than 0, we need to solve the inequality $x^2 - 4 < 0$ .

First, find when the function equals zero by solving $x^2 - 4 = 0$ . This gives the roots as $x^2 = 4$ , or $x = \pm 2$ , specifically $x = 2$ and $x = -2$ .

Next, perform a sign analysis of $f(x) = x^2 - 4$ in the intervals defined by these roots: $(-\infty, -2)$ , $(-2, 2)$ , and $(2, \infty)$ .

For $x \in (-\infty, -2)$ : Pick a test point, such as $x = -3$ . Calculate $f(-3) = (-3)^2 - 4 = 9 - 4 = 5$ . Here, the function is positive.
For $x \in (-2, 2)$ : Pick a test point, such as $x = 0$ . Calculate $f(0) = 0^2 - 4 = -4$ . Here, the function is negative, satisfying $f(x) < 0$ .
For $x \in (2, \infty)$ : Pick a test point, such as $x = 3$ . Calculate $f(3) = 3^2 - 4 = 9 - 4 = 5$ . Here, the function is positive.

Thus, the function $f(x) = x^2 - 4$ is less than 0 for $-2 < x < 2$ .

The correct interval representing the solution is $-2 < x < 2$ .

Final Answer

$-2 < x < 2$

Key Points to Remember

Essential concepts to master this topic

Rule: Find zeros first by solving $x^2 - 4 = 0$
Technique: Test points in intervals: $f(0) = 0^2 - 4 = -4 < 0$
Check: Verify boundary points excluded: at $x = ±2$ , function equals zero ✓

Common Mistakes

Avoid these frequent errors

Including boundary points in the solution
Don't include x = 2 and x = -2 in your answer when solving f(x) < 0 = wrong solution set! At these points, the function equals zero, not less than zero. Always use open intervals like (-2, 2) when the inequality is strictly less than.

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 do I need to find where the function equals zero first?

The zeros (where $x^2 - 4 = 0$ ) are the boundary points where the function changes from positive to negative. These critical points at $x = ±2$ divide the number line into intervals to test.

How do I know which intervals to test?

After finding zeros at $x = -2$ and $x = 2$ , test one point in each interval: before -2, between -2 and 2, and after 2. Pick easy numbers like -3, 0, and 3.

What's the difference between < and ≤ in the answer?

For $f(x) < 0$ , use open intervals like $-2 < x < 2$ (excludes endpoints). For $f(x) ≤ 0$ , use closed intervals like $-2 ≤ x ≤ 2$ (includes endpoints).

Can I just look at the graph instead of doing algebra?

Yes! The graph shows where the parabola is below the x-axis (negative values). But always verify algebraically by finding the exact zeros and testing intervals for complete accuracy.

Why is the answer an interval instead of specific values?

Unlike equations that have specific solutions, inequalities have ranges of solutions. Every x-value between -2 and 2 makes the function negative, so the solution is the entire interval.

What if I get a different interval when testing?

Double-check your zeros by solving $x^2 - 4 = 0$ correctly. Then carefully substitute your test points: $f(0) = -4$ (negative) confirms the middle interval is part of the solution.

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