Problem 17 For Exercises 7 to $75,$ facto... [FREE SOLUTION]

Chapter 10: Problem 17

For Exercises 7 to $75,$ factor. $$b^{2}-6 b+9$$

Short Answer

Expert verified

The factorized form of the given expression $b^{2}-6b+9$ is $(b - 3)^2$.

Step by step solution

Identify the coefficients

The given expression is $b^{2}-6b+9$. Here a = 1 is the coefficient of $b^{2}$, b = -6 is the coefficient of b, and c = 9 is the constant term.

Check for perfect square

A quadratic expression $ax^{2}+bx+c$ is a perfect square if the constant term c is equal to $(b/2a)^{2}$. Here, $(-6/2(1))^{2} = (-3)^{2} = 9$ which is equal to c.

Factoring the Expression

Since it is a perfect square, the expression can be factored as $(b - 3)^{2}$.This is because, in the given equation the term -3 is half of the b term -6, and the square of -3 gives the constant term 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic expression that can be written as the square of a binomial. This happens when the quadratic takes the form $a^2 - 2ab + b^2 = (a - b)^2$ or $a^2 + 2ab + b^2 = (a + b)^2$. Identifying them makes factoring simpler and faster.
For the quadratic expression $b^2 - 6b + 9$:

The first term $b^2$ is a perfect square of $b$.
The last term $9$ is a perfect square of $3$.
The middle term $-6b$ is twice the product of $b$ and $3$ (i.e., $-2 \times b \times 3$).

Recognizing all these parts confirms the expression is a perfect square trinomial, specifically $(b - 3)^2$.
Understanding this concept can make it quicker to factor without exhaustive trial and error.

Quadratic Expressions

Quadratic expressions follow a general form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. These expressions graph as parabolas and often require factoring to solve or simplify.
In the example $b^2 - 6b + 9$:

The coefficient of $b^2$ is 1, making it a simple quadratic to factor.
The coefficient of $b$ is $-6$, which affects the parabola's direction and axis.
The constant term is 9, impacting the vertical position on the graph.

Each term has a role, and understanding these helps in manipulating and factoring quadratics.

Factoring Techniques

Factoring quadratics can employ several techniques, depending on their form. For perfect square trinomials, the process is straightforward. However, other quadratics might require:

Trial and Error: Testing factor pairs of the constant term and checking sums or products.
Grouping: Splitting the middle term for grouping and factor extraction.
Quadratic Formula: For complex expressions, using $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

For $b^2 - 6b + 9$, because it's a perfect square trinomial, recognize the square root relationships and factor it directly into $(b - 3)^2$. With practice, identifying the best approach becomes faster.

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91影视

For Exercises 7 to \(75,\) factor. $$b^{2}-6 b+9$$

Short Answer

Step by step solution

Identify the coefficients

Check for perfect square

Factoring the Expression

Key Concepts

Perfect Square Trinomial

Quadratic Expressions

Factoring Techniques

One App. One Place for Learning.

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