Questions: Factor the following. 25y^2 + 10y + 1 - x^2

Transcript text: Factor the following. \[ 25 y^{2}+10 y+1-x^{2} \]

Solution

Solution Steps

To factor the given expression \( 25y^2 + 10y + 1 - x^2 \), we can recognize it as a difference of squares. The expression can be rewritten as \((5y + 1)^2 - x^2\), which fits the form \(a^2 - b^2\). This can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\).

Step 1: Rewrite the Expression

We start with the expression: \[ 25y^2 + 10y + 1 - x^2 \] This can be rearranged as: \[ -x^2 + 25y^2 + 10y + 1 \]

Step 2: Factor the Expression

Recognizing that the expression can be factored as a difference of squares, we rewrite it in the form: \[ -(x^2 - (5y + 1)^2) \] Using the difference of squares identity \(a^2 - b^2 = (a - b)(a + b)\), we have: \[ -(x - (5y + 1))(x + (5y + 1)) \]

Step 3: Final Factored Form

Thus, the complete factored form of the expression is: \[ -(x - 5y - 1)(x + 5y + 1) \]