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

Video Solution

Solution Steps

00:00 Find the positive domain of the function

00:03 Looking at the coefficient of X squared, which is positive, therefore the function is smiling

00:11 The positive domain is actually above the X-axis

00:14 Therefore, we'll set Y=0 to find the intersection points with the X-axis

00:21 Isolate X

00:32 Extract the root

00:38 When extracting a root, there are 2 solutions (positive and negative)

00:43 These are the intersection points with the X-axis

00:56 Let's mark the intersection points with the X-axis

01:06 The positive domain is above the X-axis

01:10 The negative domain is below the X-axis

01:14 Let's find the domain where the function is positive

01:21 And this is the solution to the question

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 $5 < x$ .