Calculate Positive Area: Finding Area of f(x)=2x²-50

Q: Why do I need to find the roots first if I'm looking for positive areas?

The roots are boundary points where the function changes sign! For $ f(x) = 2x^2 - 50 $, the roots x = ±5 divide the number line into intervals where the function is consistently positive or negative.

Q: How do I know which test values to pick in each interval?

Pick any convenient number in each interval! For $ x , try x = -6. For \( -5 , try x = 0. For \( x > 5 $, try x = 6. The specific value doesn't matter as long as it's in the right interval.

Q: What does 'positive area' actually mean?

Positive area means where the function is above the x-axis (f(x) > 0). We're finding the x-values where the parabola is positive, not calculating an actual area measurement.

Q: Why is the middle interval negative when it's between the roots?

This parabola $ f(x) = 2x^2 - 50 $ opens upward (positive coefficient of x²). Between the roots, it dips below the x-axis, making f(x) negative. Outside the roots, it rises above the x-axis, making f(x) positive.

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

Graphing helps visualize, but algebraic methods give exact answers! Testing intervals ensures you get the precise boundaries ($ x or \( x > 5 $) rather than approximate values from a sketch.

Quadratic Inequalities with Sign Analysis

Find the positive area of the function

$f(x)=2x^2-50$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find where the function is positive.

00:10 The coefficient of X squared is positive. This means our parabola smiles, or opens upwards.

00:18 So, the positive domain is where the graph is above the X axis.

00:22 To find intersections with the X axis, we'll set Y equal to zero.

00:28 Now, let's solve for X.

00:39 To do this, we'll extract the square root.

00:45 Remember, extracting a root gives us two solutions; one positive and one negative.

00:50 These solutions are the points where our graph touches the X axis.

01:03 Let's mark these intersection points on the graph.

01:13 The sections above the X axis are the positive domain.

01:18 Below the X axis, we find the negative domain.

01:21 Now, we'll focus on the domain where our function is positive.

01:28 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive area of the function

$f(x)=2x^2-50$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Solve the equation $2x^2 - 50 = 0$ to find the roots.
Step 2: Analyze intervals on the number line based on the roots to find where $f(x) > 0$ .

Now, let's work through each step:
Step 1: Set the function to zero: $2x^2 - 50 = 0$ . Solving gives:

2x^2 = 50 \\ x^2 = 25 \\ x = \pm 5

Step 2: The roots divide the number line into intervals: $x < -5$ , $-5 < x < 5$ , and $x > 5$ . We test these intervals in $f(x) = 2x^2 - 50$ .

For $x < -5$ , pick $x = -6$ : $f(-6) = 2(-6)^2 - 50 = 72 - 50 = 22 > 0$ This interval is positive.
For $-5 < x < 5$ , pick $x = 0$ : $f(0) = 2(0)^2 - 50 = -50 < 0$ This interval is negative.
For $x > 5$ , pick $x = 6$ : $f(6) = 2(6)^2 - 50 = 72 - 50 = 22 > 0$ This interval is positive.

Therefore, the function is positive for $x < -5$ and $x > 5$ .

Hence, the solution to the problem is $x < -5$ or $5 < x$ .

Final Answer

$x < -5$ o $5 < x$

Key Points to Remember

Essential concepts to master this topic

Root Finding: Set quadratic equal to zero to find boundary points
Interval Testing: Pick test values like x = -6, 0, 6 in each interval
Verification: Check that f(-6) = 22 > 0 and f(6) = 22 > 0 ✓

Common Mistakes

Avoid these frequent errors

Testing only one interval or guessing the sign pattern
Don't assume the parabola is positive everywhere outside the roots without testing = wrong intervals! The sign can change unpredictably. Always test a specific value in each interval created by the roots.

Practice Quiz

Test your knowledge with interactive questions

Find the ascending area of the function

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

FAQ

Everything you need to know about this question

Why do I need to find the roots first if I'm looking for positive areas?

The roots are boundary points where the function changes sign! For $f(x) = 2x^2 - 50$ , the roots x = ±5 divide the number line into intervals where the function is consistently positive or negative.

How do I know which test values to pick in each interval?

Pick any convenient number in each interval! For $x < -5$ , try x = -6. For $-5 < x < 5$ , try x = 0. For $x > 5$ , try x = 6. The specific value doesn't matter as long as it's in the right interval.

What does 'positive area' actually mean?

Positive area means where the function is above the x-axis (f(x) > 0). We're finding the x-values where the parabola is positive, not calculating an actual area measurement.

Why is the middle interval negative when it's between the roots?

This parabola $f(x) = 2x^2 - 50$ opens upward (positive coefficient of x²). Between the roots, it dips below the x-axis, making f(x) negative. Outside the roots, it rises above the x-axis, making f(x) positive.

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

Graphing helps visualize, but algebraic methods give exact answers! Testing intervals ensures you get the precise boundaries ( $x < -5$ or $x > 5$ ) rather than approximate values from a sketch.

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