Ask AI: factor b²+16b+64

Sure! To factor the quadratic expression \( b^2 + 16b + 64 \), we start by checking if it can be written as the square of a binomial. Notice that \( 64 \) is a perfect square (\( 8^2 = 64 \)), and \( 16b \) can be seen as \( 2 \cdot 8 \cdot b \). These observations suggest that the quadratic may be a perfect square trinomial. A perfect square trinomial can be written as: \[ (b + d)^2 \] where \( d \) is a number such that \( d^2 = 64 \) and \( 2bd = 16b \). In this case: \[ d = 8 \] Thus, the expression can be factored as: \[ (b + 8)^2 \] We can verify this by expanding \( (b + 8)^2 \): \[ (b + 8)(b + 8) = b^2 + 8b + 8b + 64 = b^2 + 16b + 64 \] So, the factored form of \( b^2 + 16b + 64 \) is: \[ (b + 8)^2 \]