Problem 292
Question
Factor completely. \(49 b^{2}-112 b+64\)
Step-by-Step Solution
Verified Answer
\( (7b-8)^2 \)
1Step 1: Identify the quadratic expression
The given expression is a quadratic polynomial: \[49b^2 - 112b + 64\]
2Step 2: Recognize the perfect square trinomial form
Compare the quadratic expression to the form \(a^2 - 2ab + b^2\).The given expression \(49b^2 - 112b + 64\) can be written as \( (7b)^2 - 2(7b)(8) + 8^2\).Thus, the polynomial is a perfect square trinomial.
3Step 3: Write the quadratic expression as a squared binomial
Since \( 49b^2 - 112b + 64 \) is a perfect square trinomial, it simplifies to \( (7b - 8)^2 \).
4Step 4: Verify the factorization
Expand \( (7b - 8)^2 \) to ensure the expression equals the original quadratic polynomial. \((7b - 8)^2 = (7b - 8)(7b - 8)\) \[ (7b)(7b) - 8(7b) - 8(7b) + 64 = 49b^2 - 56b - 56b + 64 = 49b^2 - 112b + 64\] The factorization is correct.
Key Concepts
Perfect Square TrinomialQuadratic ExpressionBinomial
Perfect Square Trinomial
A perfect square trinomial is a special type of quadratic expression. It can be written in the form \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). These expressions are called perfect square trinomials because they can be factored into the square of a binomial.
For example, consider the expression \[ 49b^2 - 112b + 64 \]. By observing the coefficients and constants, we can rewrite it as \( (7b)^2 - 2(7b)(8) + 8^2 \). This is the standard form of a perfect square trinomial.
Identifying and rewriting the given quadratic expression in this form makes it easier to factor. You’ll see that it simplifies to \( (7b - 8)^2 \). This step is crucial, as recognizing the pattern helps in quick and accurate factorization.
For example, consider the expression \[ 49b^2 - 112b + 64 \]. By observing the coefficients and constants, we can rewrite it as \( (7b)^2 - 2(7b)(8) + 8^2 \). This is the standard form of a perfect square trinomial.
Identifying and rewriting the given quadratic expression in this form makes it easier to factor. You’ll see that it simplifies to \( (7b - 8)^2 \). This step is crucial, as recognizing the pattern helps in quick and accurate factorization.
Quadratic Expression
A quadratic expression is a polynomial of degree two. It usually takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is an unknown variable.
In the given exercise, \[49b^2 - 112b + 64\], we have a quadratic expression in terms of \(b\). Here, \( a = 49 \), \( b = -112 \), and \( c = 64 \). Identifying this structure helps in various algebraic techniques, like factoring, completing the square, or using the quadratic formula.
Quadratic expressions play a significant role in algebra due to their applications in various real-world problems, such as projectile motion, area optimization, and more.
In the given exercise, \[49b^2 - 112b + 64\], we have a quadratic expression in terms of \(b\). Here, \( a = 49 \), \( b = -112 \), and \( c = 64 \). Identifying this structure helps in various algebraic techniques, like factoring, completing the square, or using the quadratic formula.
Quadratic expressions play a significant role in algebra due to their applications in various real-world problems, such as projectile motion, area optimization, and more.
Binomial
A binomial is a polynomial expression with exactly two terms. Example forms include \(a + b\) or \(a - b\).
When factoring quadratic expressions, our goal is often to rewrite them as the product of binomials. In the exercise, we factored the quadratic expression \(49b^2 - 112b + 64\) as a squared binomial: \( (7b - 8)^2 \).
Recognizing binomials within quadratic expressions helps simplify complex algebraic problems. In this case, once you see \(49b^2 - 112b + 64\) as a perfect square trinomial, you can form the binomial \(7b - 8\) and write the quadratic as \( (7b - 8)^2 \). This demonstrates the power of binomial factorization in algebra.
When factoring quadratic expressions, our goal is often to rewrite them as the product of binomials. In the exercise, we factored the quadratic expression \(49b^2 - 112b + 64\) as a squared binomial: \( (7b - 8)^2 \).
Recognizing binomials within quadratic expressions helps simplify complex algebraic problems. In this case, once you see \(49b^2 - 112b + 64\) as a perfect square trinomial, you can form the binomial \(7b - 8\) and write the quadratic as \( (7b - 8)^2 \). This demonstrates the power of binomial factorization in algebra.