Problem 90 - Algebra 2

Question

Factor each expression completely. $$ -x^{2}-6 x-9 $$

Step-by-Step Solution

Answer

The complete factoring of the expression \(-x^2-6x-9\) is \(-1(x+3)^2\).

Step 1: Identify the coefficients

Identify the coefficients of the quadratic expression. In the given expression \(-x^2-6x-9\), the coefficient of \(x^2\) (a) is -1, the coefficient of \(x\) (b) is -6, and the constant term (c) is -9.

Step 2: Rewrite the Expression

Since the coefficient of the x^2 term is negative, rewrite the expression as \(-1(x^2+6x+9)\). This gives the expression a more standard look of a quadratic trinomial where the coefficient of the x^2 term is positive.

Step 3: Check for 'Perfect Square Trinomial'

A quadratic trinomial \(x^2+2px+p^2\) is a perfect square trinomial if it can be written as \((x+p)^2\). In our case, we can see that evaulating \(p = 3\) to fit into our trinomial matches it. Thus the expression inside the brackets is a perfect square trinomial and can be written as \((x+3)^2\).

Step 4: Write Final Result

Putting all this together, the final factored form of \(-x^2-6x-9\) is \(-1(x+3)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be expressed in the form \((x + p)^2\). This form implies that both factors are identical. Recognizing this pattern can simplify factoring greatly.

• The general structure of a perfect square trinomial is \(x^2 + 2px + p^2\).
• The middle term is twice the product of the root of the first and last terms. It follows the formula \(2px\).
• Knowing the pattern allows us to factor the expression quickly as \((x+p)^2\).

In the example provided, we modified the expression \(-x^2 - 6x - 9\) to \(x^2 + 6x + 9\), and identified it as a perfect square trinomial. This let us rewrite it in the factored form \((x + 3)^2\). Because the overall expression was multiplied by -1 originally, the complete factorization becomes \(-1(x + 3)^2\).

Coefficients in Quadratic Expressions

In any quadratic expression formatted as \(ax^2 + bx + c\), the coefficients serve as the numbers in front of the variable terms. They play a crucial role in shaping the behavior of the expression and in determining its factorization.

• a is the coefficient of \(x^2\) and dictates the opening direction and width of the parabola.
• b is the coefficient of \(x\) and influences the slope and position of the parabola.
• c is the constant term and represents the y-intercept of the function when plotted on a graph.

In our expression, the coefficients are identified as follows:

• a = -1
• b = -6
• c = -9

Because the coefficient \(a = -1\) is negative, the expression was rewritten by factoring \(-1\) to give us a standard quadratic expression, making it easier to identify and factor with a positive leading term.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms in the expression to reveal common factors. It is frequently used when direct factoring methods are challenging. In the context of quadratic expressions, this method can sometimes help when standard techniques like recognizing perfect squares aren't possible or straightforward.

• First, arrange the expression to separate the terms in a meaningful way.
• Next, look for a common factor within each group.
• Finally, if successful, factor out these commonalities, which often simplifies to a clear factored form.

In the exercise, while we didn't directly employ grouping, preparing the expression by factoring out \(-1\) simulates a similar strategic step by reforming it from \(-x^2 - 6x - 9\) to \(x^2 + 6x + 9\). This step made it simpler to apply the recognition of the perfect square trinomial.