Question

Factor the perfect squares. $$ 16 x^{2}+8 x+1 $$

Short answer

(4x + 1)^{2}

Step-by-step solution

Identify the form of the quadratic expression

Notice that the given expression is in the form of a perfect square trinomial. A perfect square trinomial has the form \[a^{2} + 2ab + b^{2}\].

Relate with the perfect square trinomial form

Compare \[16x^{2} + 8x + 1\] with \[a^{2} + 2ab + b^{2}\]. We can see that \[16x^{2}\] corresponds to \[a^{2}\], \[8x\] corresponds to \[2ab\], and \[1\] corresponds to \[b^{2}\].

Find values of a and b

From \[16x^{2} = a^{2}\], solve for \[a\]. We get \[a = 4x\]. From \[1 = b^{2}\], solve for \[b\]. We get \[b = 1\].

Verify the middle term

Ensure that \[2ab\] (\[2 \times 4x \times 1 = 8x\]) matches the middle term of the given expression.

Write the expression as a squared binomial

Since all terms match perfectly, write the original expression \[16x^{2} + 8x + 1\] as \[(4x + 1)^{2}\].

Key concepts

Factoring Quadratics

Factoring quadratics involves breaking down quadratic expressions into simpler expressions that, when multiplied together, give the original quadratic. This process is essential in solving quadratic equations and simplifying expressions. For example, given the quadratic expression \(16x^{2}+8x+1\), our goal is to find factors that multiply to the same result.
In this exercise, the expression is already a perfect square trinomial, making it easier to factor. Let's see how we can approach it step-by-step.

Quadratic Expressions

Understanding quadratic expressions is crucial. A quadratic expression generally has the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants. The given expression in our exercise, \(16x^2 + 8x + 1\), is a type of quadratic expression because it fits the general form where \(a = 16, b = 8,\) and \(c = 1\).
Our particular focus here is on perfect square trinomials, a special form of quadratic expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging expressions and equations using algebraic rules. The objective is to simplify or solve for variables. In the context of our exercise, successful algebraic manipulation helps us identify the components of the perfect square trinomial \(a^2 + 2ab + b^2\) form.
By comparing \(16x^2 + 8x + 1\) with \(a^2 + 2ab + b^2\), we manipulated the algebraic terms to match the standard form, aiding in efficient factoring.

Binomial Squares

A binomial square is an expression of the form \((a + b)^2\) or \((a - b)^2\). When expanded, it forms a perfect square trinomial: \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). The goal of our exercise is to express \(16x^2 + 8x + 1\) as a binomial square.
Through steps involving identifying \a\ and \b\, and verifying the middle term, we concluded that \((4x + 1)^2\) is the binomial square representation. This is a critical skill in algebraic problem-solving, simplifying complex quadratic equations, and enhancing mathematical fluency.