Compare Square Expressions: (x-4)² vs x²+16 with x>0

Question

Fill in the corresponding sign given that

x > 0

(x4)2?x2+16 (x-4)^2?x^2+16

Video Solution

Solution Steps

00:00 Complete the appropriate sign
00:04 We'll use shortened multiplication formulas to open the parentheses
00:22 Let's solve the multiplication
00:31 Let's reduce what we can
00:39 X is positive, therefore the entire expression is negative - less than 0
00:46 And this is the solution to the question

Step-by-Step Solution

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

  • Step 1: Expand both expressions.
  • Step 2: Subtract one expression from the other to determine the inequality.
  • Step 3: Analyze and conclude based on the comparison.

Now, let's work through each step:
Step 1: Expand (x4)2(x-4)^2:

(x4)2=(x4)(x4)=x224x+16=x28x+16(x-4)^2 = (x-4)(x-4) = x^2 - 2 \cdot 4 \cdot x + 16 = x^2 - 8x + 16

Step 2: Set up the inequality (x4)2<x2+16(x-4)^2 < x^2 + 16 and substitute the expanded form:

x28x+16<x2+16x^2 - 8x + 16 < x^2 + 16

Step 3: Simplify the inequality by subtracting x2x^2 and 1616 from both sides:

x28x+16x216<0x^2 - 8x + 16 - x^2 - 16 < 0

8x<0-8x < 0

Solving 8x<0-8x < 0 gives:

x>0x > 0

This inequality holds true for all x>0x > 0.

Therefore, the inequality (x4)2<x2+16(x-4)^2 < x^2 + 16 is correct for x>0x > 0.

Thus, the correct symbol to fill in the blank is <\lt.

Answer

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