Problem 10 - Elementary and Intermediate Algebra

Question

What is the result of the subtraction in the \(x\) -column? $$ \begin{array}{rl} {8 x^{2}-7 x-1} & {8 x^{2}-7 x-1} \\ {-\left(4 x^{2}+6 x-9\right)} & {-4 x^{2}-6 x+9} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result is 4x^{2} - x - 10 .

Step 1: Align the Terms

First, align the like terms vertically. You have the quadratic terms ( 8x^{2} and -4x^{2} ), the linear terms ( -7x and -6x ), and the constant terms ( -1 and 9 ).

Step 2: Subtract the Quadratic Terms

Subtract the quadratic terms: 8x^{2} - 4x^{2} = 4x^{2} .

Step 3: Subtract the Linear Terms

Subtract the linear terms: -7x - (-6x) = -7x + 6x = -x . Remember, subtracting a negative is equivalent to adding.

Step 4: Subtract the Constant Terms

Subtract the constants: -1 - 9 = -10 . You simply subtract the second constant from the first.

Step 5: Combine the Results

Combine the results from Steps 2, 3, and 4 to form the final expression from the subtraction: 4x^{2} - x - 10 .

Key Concepts

Quadratic Terms

In mathematics, "quadratic terms" refer to expressions involving a variable raised to the power of two. These terms form a crucial part of quadratic equations. In the given problem, the quadratic terms are given by \(8x^2\) and \(-4x^2\). These terms appear at the highest degree in a given expression, usually represented as \(ax^2\), where \(a\) is a coefficient and \(x\) is the variable.
This coefficient can be any real number, positive, negative, or zero. The term "quadratic" derives from "quadratus," the Latin word for "square," since the variable is squared in these expressions.

- To subtract quadratic terms like \(8x^2 - 4x^2\), line up similar terms and subtract their coefficients: - Identify coefficients: \(8\) and \(-4\). - Subtract coefficients: \(8 - 4 = 4\). - Combine result with \(x^2\): \(4x^2\).
This process simplifies the expression, combining all like terms into one quadratic term in the result.

Linear Terms

Linear terms have a variable raised to the power of one. They are represented in the form \(bx\), where \(b\) is the coefficient. These terms correspond to the linear part of an equation which affects the degree of slope or direction of a graph.
In the exercise, the linear terms involved are \(-7x\) and \(-6x\). When subtracting linear terms, treat the problem like a regular subtraction of numbers, after taking note of the signs, especially negative ones.

• Understand potential reactions like: when subtracting a negative, switch to addition.
• Perform the subtraction: \(-7x - (-6x) = -7x + 6x\).
• Solve it step by step: \(-7 + 6 = -1\).
• Thus, the resulting linear term is \(-x\).

Linear terms are significant because they indicate directionality in functions, affecting how graphs curve or slope.

Constant Terms

Constant terms are the standalone numbers without any variables attached. In other words, they're plain numbers either added or subtracted from the variable terms in an equation.
For the given problem, the constant terms are \(-1\) and \(9\). These represent the basic arithmetic part of subtraction, not bound by variables.

- Handling constant term subtraction: - Line up terms: look at \(-1\) and \(9\). - Perform arithmetic subtraction: \(-1 - 9\). - Calculate result: \(-1 - 9 = -10\).
Constant terms help define the starting point when graphing an equation or simplifying mathematical solutions. Even without variables, understanding their role is crucial to interpreting the full outcome of expressions correctly.