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Chapter 5: Problem 92

Perform the operations. $$ (4 a-3)^{2}+(a+6)^{2} $$

Short Answer

Expert verified
The simplified expression is \(17a^2 - 12a + 45\).

Step by step solution

01

Expand the First Expression

Start by expanding the expression \((4a - 3)^2\). Use the formula for squaring a binomial, which is \((x-y)^2 = x^2 - 2xy + y^2\). Here, let \(x = 4a\) and \(y = 3\). Therefore, \((4a - 3)^2 = (4a)^2 - 2(4a)(3) + 3^2 = 16a^2 - 24a + 9\).
02

Expand the Second Expression

Next, expand the expression \((a + 6)^2\) using the formula \((x + y)^2 = x^2 + 2xy + y^2\). Here, \(x = a\) and \(y = 6\). The expansion is \(a^2 + 2(a)(6) + 6^2 = a^2 + 12a + 36\).
03

Add the Expanded Expressions

Now add the two expanded expressions: \(16a^2 - 24a + 9\) and \(a^2 + 12a + 36\). Combine like terms to simplify: \(16a^2 + a^2 = 17a^2\), \(-24a + 12a = -12a\), and \(9 + 36 = 45\). Thus, the simplified expression is \(17a^2 - 12a + 45\).

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Key Concepts

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

Binomial Expansion
Binomial expansion is a process used to expand expressions that are raised to a power. In our exercise, we used this method to expand the binomials \((4a - 3)^2\) and \((a + 6)^2\). The general formula for expanding the square of a binomial \((x - y)^2\) is:
  • First, square the first term: \(x^2\)
  • Second, multiply the two terms together, double it: \(-2xy\)
  • Third, square the second term: \(y^2\)
Combine these results to form the expanded expression: \(x^2 - 2xy + y^2\).
For example, in our first expression \((4a - 3)^2\):
  • Square \(4a\), which gives \(16a^2\).
  • Multiply \(4a\) and \(-3\), double it to get \(-24a\).
  • Finally, square \(-3\) to obtain \(9\).
Similarly, you apply the process with \((x + y)^2\) to expand \((a + 6)^2\). This step-by-step expansion helps us transform a binomial into a polynomial form.
Polynomials
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. They often appear as the sum of monomials, like \(a^2 + 12a + 36\). In the exercise, we dealt with transforming the binomials into polynomials by expanding them.
These formed polynomials:
  • From \((4a - 3)^2\), we get \(16a^2 - 24a + 9\).
  • From \((a + 6)^2\), we have \(a^2 + 12a + 36\).
Polynomials are fundamental in algebra as they can be used to model a wide range of problems. They represent equations that allow us to perform various operations, such as addition, subtraction, and multiplication. Understanding polynomials involves recognizing the parts of each term:\( ax^n\) where \(a\) is the coefficient and \(n\) is the exponent, indicating the degree of the polynomial.
Combining Like Terms
Combining like terms is an essential step in simplifying polynomials. In the final step of our exercise, we merged the terms of the two expanded polynomials. Each of these terms has parts:\
  • The same variable letter: \(a\)
    • \
  • The same exponent: 2 and 1 in our case
    • Matching terms make it easier to simplify an expression effectively. In the given problem:
    • Add \(16a^2 + a^2\) to get \(17a^2\)
    • Combine \(-24a + 12a\) resulting in \(-12a\)
    • Add constants \(9 + 36\) to achieve \(45\)
    When you simplify using like terms, the expression is neater and easier to interpret. This method of combining terms helps in problem-solving by reducing polynomial expressions to their simplest form.

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