Factor. 16z^2 - 8z + 1 ______

Identify Quadratic Form: Determine if the quadratic can be factored using the form (az - b)^2 = a^2z^2 - 2abz + b^2. We notice that 16z^2 is a perfect square, as is 1. The middle term is -8z, which is twice the product of the square roots of 16z^2 and 1, with a negative sign. This suggests that the quadratic is a perfect square trinomial.Find Square Roots: Identify the square roots of the first and last terms. The square root of 16z^2 is 4z, and the square root of 1 is 1. Write in (az - b)^2 Form: Write the quadratic in the form of (az - b)^2 using the identified square roots. The quadratic 16z^2 - 8z + 1 can be written as (4z - 1)^2 because (4z)^2 = 16z^2, -2*(4z)*1 = -8z, and 1^2 = 1.Factor the Quadratic: Factor the quadratic expression. Since we have identified the quadratic as a perfect square trinomial, we can write it as the square of a binomial. Therefore, the factored form of 16z^2 - 8z + 1 is (4z - 1)^2.

Home

Math Problems

Algebra 1

Factor quadratics: special cases

Full solution

Q. Factor.

\newline

16z^2 - 8z + 1

Identify Quadratic Form: Determine if the quadratic can be factored using the form $(az - b)^2 = a^2z^2 - 2abz + b^2$ . We notice that $16z^2$ is a perfect square, as is $1$ . The middle term is $-8z$ , which is twice the product of the square roots of $16z^2$ and $1$ , with a negative sign. This suggests that the quadratic is a perfect square trinomial.
Find Square Roots: Identify the square roots of the first and last terms. $\newline$ The square root of $16z^2$ is $4z$ , and the square root of $1$ is $1$ .
Write in $(az - b)^2$ Form: Write the quadratic in the form of $(az - b)^2$ using the identified square roots. $\newline$ The quadratic $16z^2 - 8z + 1$ can be written as $(4z - 1)^2$ because $(4z)^2 = 16z^2$ , $-2*(4z)*1 = -8z$ , and $1^2 = 1$ .
Factor the Quadratic: Factor the quadratic expression. $\newline$ Since we have identified the quadratic as a perfect square trinomial, we can write it as the square of a binomial. Therefore, the factored form of $16z^2 - 8z + 1$ is $(4z - 1)^2$ .