What Is The Product Of $(4x - 7)(4x + 7)$?A. $16x^2 + 56x - 49$B. \$8x^2 - 49$[/tex\]C. $8x^2 + 56x - 49$D. $16x^2 - 49$
Understanding the Problem
When dealing with algebraic expressions, one common operation is multiplication. In this case, we are given the expression (4x - 7)(4x + 7) and asked to find its product. To solve this problem, we will use the distributive property, which allows us to multiply each term in the first expression by each term in the second expression.
Using the Distributive Property
The distributive property states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
We can apply this property to our given expression by multiplying each term in the first expression by each term in the second expression.
(4x - 7)(4x + 7) = (4x)(4x) + (4x)(7) - (7)(4x) - (7)(7)
Simplifying the Expression
Now that we have applied the distributive property, we can simplify the expression by combining like terms.
(4x)(4x) = 16x^2 (4x)(7) = 28x -(7)(4x) = -28x -(7)(7) = -49
Combining Like Terms
Now that we have simplified the expression, we can combine like terms to get the final result.
16x^2 + 28x - 28x - 49
Canceling Out Like Terms
Notice that the 28x and -28x terms cancel each other out, leaving us with:
16x^2 - 49
Conclusion
Therefore, the product of (4x - 7)(4x + 7) is 16x^2 - 49.
Comparison with Answer Choices
Let's compare our result with the answer choices:
A. 16x^2 + 56x - 49 B. 8x^2 - 49 C. 8x^2 + 56x - 49 D. 16x^2 - 49
Our result matches answer choice D.
Final Answer
The final answer is D. 16x^2 - 49.
Additional Tips and Tricks
When dealing with algebraic expressions, it's essential to remember the following tips and tricks:
- Use the distributive property to multiply expressions.
- Simplify expressions by combining like terms.
- Cancel out like terms to get the final result.
- Compare your result with answer choices to ensure accuracy.
By following these tips and tricks, you can become more confident and proficient in solving algebraic expressions.
Q: What is the distributive property in algebra?
A: The distributive property is a fundamental concept in algebra that allows us to multiply each term in one expression by each term in another expression. It states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
Q: How do I apply the distributive property to an expression?
A: To apply the distributive property, simply multiply each term in the first expression by each term in the second expression. For example, if we have the expression (4x - 7)(4x + 7), we would multiply each term in the first expression by each term in the second expression:
(4x - 7)(4x + 7) = (4x)(4x) + (4x)(7) - (7)(4x) - (7)(7)
Q: What is the difference between like terms and unlike terms?
A: Like terms are terms that have the same variable(s) raised to the same power. Unlike terms are terms that have different variables or variables raised to different powers. For example, 2x and 3x are like terms, while 2x and 5y are unlike terms.
Q: How do I simplify an expression by combining like terms?
A: To simplify an expression by combining like terms, simply add or subtract the coefficients of like terms. For example, if we have the expression 2x + 3x, we would combine the like terms by adding their coefficients:
2x + 3x = (2 + 3)x = 5x
Q: What is the order of operations in algebra?
A: The order of operations in algebra is a set of rules that tells us which operations to perform first when working with expressions that involve multiple operations. The order of operations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I evaluate an expression with multiple operations?
A: To evaluate an expression with multiple operations, simply follow the order of operations. For example, if we have the expression 2x + 3(4x - 2), we would follow the order of operations as follows:
- Evaluate the expression inside the parentheses: 4x - 2 = 4x - 2
- Multiply 3 by the result: 3(4x - 2) = 12x - 6
- Add 2x to the result: 2x + 12x - 6 = 14x - 6
Q: What is the difference between an expression and an equation?
A: An expression is a group of variables, constants, and mathematical operations that are combined using the rules of algebra. An equation is a statement that says two expressions are equal. For example, 2x + 3 = 5 is an equation, while 2x + 3 is an expression.
Q: How do I solve an equation?
A: To solve an equation, simply isolate the variable by performing the necessary operations to get the variable by itself on one side of the equation. For example, if we have the equation 2x + 3 = 5, we would solve for x by subtracting 3 from both sides and then dividing both sides by 2:
2x + 3 = 5 2x = 5 - 3 2x = 2 x = 2/2 x = 1
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation is an equation in which the highest power of the variable is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 2x + 1 = 0 is a quadratic equation.
Q: How do I solve a quadratic equation?
A: To solve a quadratic equation, simply use the quadratic formula:
x = (-b ± â(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. For example, if we have the quadratic equation x^2 + 2x + 1 = 0, we would use the quadratic formula to solve for x:
x = (-2 ± â(2^2 - 4(1)(1))) / 2(1) x = (-2 ± â(4 - 4)) / 2 x = (-2 ± â0) / 2 x = (-2 ± 0) / 2 x = -2/2 x = -1
Q: What is the difference between a rational expression and an irrational expression?
A: A rational expression is an expression that can be written as the ratio of two polynomials. An irrational expression is an expression that cannot be written as the ratio of two polynomials. For example, 2x / (x + 1) is a rational expression, while âx is an irrational expression.
Q: How do I simplify a rational expression?
A: To simplify a rational expression, simply cancel out any common factors in the numerator and denominator. For example, if we have the rational expression 2x / (x + 1), we would simplify it by canceling out the common factor x:
2x / (x + 1) = 2 / (1 + 1/x) = 2 / (1 + 1/x) Ă (x / x) = 2x / (x + 1)
Q: What is the difference between a polynomial and a non-polynomial expression?
A: A polynomial is an expression that consists of variables and constants combined using only addition, subtraction, and multiplication. A non-polynomial expression is an expression that contains any operation other than addition, subtraction, and multiplication. For example, 2x^2 + 3x - 1 is a polynomial, while 2x^2 + 3x - 1 / (x + 1) is a non-polynomial expression.
Q: How do I evaluate a polynomial expression?
A: To evaluate a polynomial expression, simply substitute the given values into the expression and perform the necessary operations. For example, if we have the polynomial expression 2x^2 + 3x - 1 and we want to evaluate it at x = 2, we would substitute x = 2 into the expression and perform the necessary operations:
2x^2 + 3x - 1 = 2(2)^2 + 3(2) - 1 = 2(4) + 6 - 1 = 8 + 6 - 1 = 13
Q: What is the difference between a function and a relation?
A: A function is a relation in which each input corresponds to exactly one output. A relation is a set of ordered pairs that satisfy a certain condition. For example, f(x) = 2x + 1 is a function, while {(1, 2), (2, 3), (3, 4)} is a relation.
Q: How do I determine if a relation is a function?
A: To determine if a relation is a function, simply check if each input corresponds to exactly one output. For example, if we have the relation {(1, 2), (2, 3), (3, 4)}, we would check if each input corresponds to exactly one output:
- Input 1 corresponds to output 2.
- Input 2 corresponds to output 3.
- Input 3 corresponds to output 4.
Since each input corresponds to exactly one output, the relation is a function.
Q: What is the difference between a domain and a range?
A: The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values. For example, if we have the function f(x) = 2x + 1, the domain is all real numbers, and the range is all real numbers greater than or equal to 1.
Q: How do I find the domain and range of a function?
A: To find the domain and range of a function, simply analyze the function and determine the set of all possible input values and the set of all possible output values. For example, if we have the function f(x) = 1 / (x - 1), the domain is all real numbers except 1, and the range is all real numbers except 0.
Q: What is the difference between a linear function and a non-linear function?
A: A linear function is a function in which the highest power of the variable is 1. A non-linear function is a function in which the highest power of the variable is greater than 1. For example, f(x) = 2x + 1 is a linear function, while f(x) = x^2 + 2x + 1 is a non-linear function.
Q: How do I determine if a function is linear or non-linear?
A: To determine if a function is linear or non-linear, simply analyze the function and determine the highest power of the variable. If the highest power is 1, the function is linear. If the highest power is greater than 1, the function is non-linear.