Question

Felix was asked whether the following equation is an identity: (x^2+1)^2=(x^2-1)^2+(2x)^2 He performed the following steps: (x^2+1)^2

Step 1 x^4+x^2+x^2+1
Step 2 x^4+2x^2+1
Step 3 x^4+2x^2+1-2x^2+2x^2 Step 4 (x^4-2x^2+1)+4x^2
Step 5 (x^2-1)^2+(2x)^2
For this reason, Felix stated that the equation is a true identity. Is Felix correct? If not, in which step did he make a mistake?

Answer 1

Felix's conclusion that the equation is a true identity is correct, but there was a mistake in Step 3 with incorrect addition and subtraction of 2x^2, which was coincidentally 'cancelled out' and did not affect the final result.

Step-by-step explanation:

The student has asked whether the following equation is an identity: (x^2+1)^2=(x^2-1)^2+(2x)^2. To verify this, we need to expand and simplify both sides of the equation and see if they are equal.

Felix's expansion of the left side is correct up to Step 2, which results in x^4+2x^2+1. However, in Step 3, Felix incorrectly adds and subtracts 2x^2 instead of 2*x^2, which should give us 4x^2, not 2x^2. The correct expansion and simplification of the left side are:

(x^2+1)^2
= x^4 + 2x^2 * 1 + 1^2
= x^4 + 2x^2 + 1

Similarly, the correct expansion of the right side is:

(x^2-1)^2 + (2x)^2
= x^4 - 2x^2 * 1 + 1^2 + 4x^2
= x^4 - 2x^2 + 1 + 4x^2
= x^4 + 2x^2 + 1

Comparing the results, we see that the two sides are indeed equal. Therefore, Felix's final conclusion that the equation is a true identity is correct, but the mistake occurs in Step 3 with the incorrect addition and subtraction of 2x^2. This does not impact the final result as the error was coincidentally 'cancelled out'.