In Exercises 49–56, factor each perfect square trinomial. x^2−14x+49

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1. Identify the structure of the trinomial. A perfect square trinomial takes the form: $a^2\; \pm \; 2ab\; +\; b^2$, where the first term is a square, the last term is a square, and the middle term is twice the product of their square roots.
2. Verify that the first term, $x^2$, is a perfect square. Its square root is $x$.
3. Check that the last term, $49$, is a perfect square. Its square root is $7$.
4. Verify the middle term, $-14x$, is equal to $2ab$, where $a\; =\; x$ and $b\; =\; 7$. Compute $2ab\; =\; 2(x)(7)\; =\; 14x$, and confirm the sign matches.
5. Write the trinomial as the square of a binomial: $(x\; -\; 7)^2$.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It takes the form a^2 ± 2ab + b^2, where a and b are real numbers. For example, x^2 - 14x + 49 can be rewritten as (x - 7)^2, since 7 is half of 14 and 49 is 7 squared.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the case of perfect square trinomials, this involves identifying the binomial that, when squared, results in the trinomial. This skill is essential for simplifying expressions and solving equations.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, typically written in the form ax^2 + bx + c, where a, b, and c are constants. Understanding the structure of quadratic expressions is crucial for recognizing patterns, such as perfect square trinomials, and for applying various methods of factoring and solving quadratic equations.