Question

Factor the perfect squares. $$ 25 x^{2}+10 x+1 $$

Short answer

\[ 25x^{2} + 10x + 1 = (5x + 1)^{2} \]

Step-by-step solution

- Identify the quadratic expression

Recognize that the quadratic expression given is a trinomial: oindent \[ 25x^{2} + 10x + 1 \]

- Recognize the perfect square form

Identify that the expression is a perfect square trinomial. A perfect square trinomial follows the form: oindent \[ a^{2} + 2ab + b^{2} = (a + b)^{2} \].Here, we compare oindent \[ 25x^{2} + 10x + 1 \] with oindent \[ a^{2} + 2ab + b^{2} \] to find values of oindent \[ a \] and oindent \[ b \].

- Determine values for a and b

Find oindent \[ a \] and oindent \[ b \] from the expression. oindent For the first term oindent \[ 25x^2 \]: oindent \[ a^2 = 25x^2 \], so oindent \[ a = 5x \]. For the last term oindent \[ 1 \]: oindent \[ b^2 = 1 \], so oindent \[ b = 1 \].

- Verify the middle term

Check the middle term oindent \[ 10x \]: oindent \[ 2ab = 2(5x)(1) = 10x \]. It matches the middle term from the original expression, confirming our values of oindent \[ a = 5x \] and oindent \[ b = 1 \].

- Write the factored form

Now write the factored form using oindent \[ a \] and oindent \[ b \] in the formula: oindent \[ (a + b)^{2} \]. Substitute oindent \[ a = 5x \] and oindent \[ b = 1 \] into the equation: oindent \[ (5x + 1)^{2} \]

Key concepts

quadratic expression

A quadratic expression is a polynomial where the highest degree is two. This means the term with the highest power of the variable is squared. Generally, it takes the form: \[ ax^2 + bx + c \]. Here, \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. Quadratic expressions can be factored, solved, and analyzed in various ways, but one common approach is factoring, especially when the expression forms a perfect square trinomial.

perfect square trinomial

A perfect square trinomial is a specific type of quadratic expression that can be written as the square of a binomial. It follows the form: \[ a^{2} + 2ab + b^{2} = (a + b)^{2} \]. Identifying perfect square trinomials is crucial because they simplify factoring significantly. To identify a perfect square trinomial, check:

• If the first and last terms are perfect squares.
• If the middle term is twice the product of the square roots of the first and last terms.

For example, in the expression \( 25x^2 + 10x + 1 \):

• First term: \( 25x^2 = (5x)^2 \)
• Last term: \( 1 = (1)^2 \)
• Middle term: \( 10x = 2 \times 5x \times 1 \)

Since all conditions match, the expression is a perfect square trinomial.

factored form

The factored form of a perfect square trinomial is achieved by rewriting it as the square of a binomial. Using the example \( 25x^2 + 10x + 1 \):

• First, recognize it as a perfect square trinomial.
• Then, use the identified values of \( a \) and \( b \).
• Substitute these into the binomial form.

In this case:

• \( a = 5x \)
• \( b = 1 \)

So, the factored form is \( (5x + 1)^2 \). Factoring simplifies the expression, making it easier to solve equations or understand the properties of the quadratic function it represents.