Question

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

Short answer

\[(x + 5)^2\]

Step-by-step solution

- Identify the Perfect Square Form

Recognize that the given expression is a quadratic trinomial. The general form for a perfect square trinomial is \[a^2 + 2ab + b^2\].

- Determine the Values of a and b

Compare the given expression \[x^2 + 10x + 25\] to the general perfect square trinomial form. Here, \[a^2 = x^2\], so \[a = x\]. The middle term tells us that \[2ab = 10x\].

- Solve for b

Given \[2ab = 10x\], substitute \[a = x\] to get \[2x \times b = 10x\]. Solving for \[b\], we have \[b = 5\]. The constant term confirms this as \[b^2 = 25\].

- Write the Factorization

Now substitute \[a = x\] and \[b = 5\] back into the perfect square form: \[(a + b)^2\]. Thus, the factorization of \[x^2 + 10x + 25\] is \[(x + 5)^2\].

Key concepts

Quadratic Trinomials

A quadratic trinomial is an expression of the form \(ax^2 + bx + c\). This type of polynomial has three terms where a\textbf{} is the coefficient of the square term, b is the coefficient of the linear term, and c is the constant term. Recognizing whether a trinomial is a perfect square can simplify the factoring process significantly. To solve and factor them, we need to understand the different forms they can take, especially when identifying perfect squares.

Perfect Square Trinomials

Perfect square trinomials have a specific structure: \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). These forms represent the squared binomials \((a + b)^2\) and \((a - b)^2\), respectively. Identifying this form involves matching the original expression to either of the patterns above. For example, in the expression \(x^2 + 10x + 25\), we compare it with \(a^2 + 2ab + b^2\). First, identify a and b: in this case, a is x (since \(x^2\text{}\) matches \(a^2\)) and b is 5 (as \(10 = 2 \times x \times 5\) and \(25 = 5^2\)). Then, rewrite the trinomial as a binomial squared: \((x + 5)^2\).

Factoring Quadratic Equations

Factoring quadratic equations, particularly when they are in the form of perfect square trinomials, simplifies problem-solving. First, recognize the trinomial's pattern—for instance, \(x^2 + 10x + 25\) can be identified as a potential perfect square. By comparing it to \(a^2 + 2ab + b^2\), find values for a and b. Verify these values using the formula \(2ab\) for the middle term, ensuring it matchesthe given expression. Then, express the trinomial as \((a + b)^2\). This process not only simplifies solving the equation but also checks the correctness of the factorization, ensuring the expression is a perfect square.