Select The Correct Answer.Which Quadratic Expression Represents The Product Of These Factors? (2x + 5)(7 - 4x)A. 8x² - 34x + 35 B. 8x + 34x - 35 C. 8x + 6x - 35 D. -8x² - 6x + 35
Understanding Quadratic Expressions
Quadratic expressions are a fundamental concept in algebra, and they play a crucial role in solving various mathematical problems. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It is typically written in the form of ax² + bx + c, where a, b, and c are constants, and x is the variable.
The Product of Factors: A Quadratic Expression
In this article, we will focus on finding the product of two quadratic factors, (2x + 5) and (7 - 4x). To do this, we need to multiply these two expressions together, using the distributive property of multiplication over addition.
Multiplying the Factors
To multiply the factors (2x + 5) and (7 - 4x), we need to follow the order of operations (PEMDAS):
- Multiply the first term of the first factor (2x) by the second term of the second factor (-4x).
- Multiply the first term of the first factor (2x) by the first term of the second factor (7).
- Multiply the second term of the first factor (5) by the second term of the second factor (-4x).
- Multiply the second term of the first factor (5) by the first term of the second factor (7).
Applying the Distributive Property
Using the distributive property, we can rewrite the multiplication of the factors as:
(2x + 5)(7 - 4x) = 2x(7) + 2x(-4x) + 5(7) + 5(-4x)
Simplifying the Expression
Now, let's simplify the expression by combining like terms:
2x(7) = 14x 2x(-4x) = -8x² 5(7) = 35 5(-4x) = -20x
Combining Like Terms
Combining the like terms, we get:
-8x² - 20x + 35
Comparing with the Options
Now, let's compare the simplified expression with the options given:
A. 8x² - 34x + 35 B. 8x + 34x - 35 C. 8x + 6x - 35 D. -8x² - 6x + 35
Selecting the Correct Answer
Based on the simplified expression, we can see that the correct answer is:
D. -8x² - 6x + 35
Conclusion
In this article, we learned how to multiply quadratic expressions by using the distributive property and simplifying the resulting expression. We also compared the simplified expression with the options given and selected the correct answer. This process is essential in solving various mathematical problems, and it requires a thorough understanding of quadratic expressions and their properties.
Tips and Tricks
- When multiplying quadratic expressions, use the distributive property to break down the multiplication into smaller steps.
- Combine like terms to simplify the expression.
- Compare the simplified expression with the options given to select the correct answer.
Real-World Applications
Quadratic expressions have numerous real-world applications, including:
- Modeling the motion of objects under the influence of gravity
- Describing the shape of a parabola
- Solving optimization problems in economics and business
Practice Problems
Try solving the following practice problems to reinforce your understanding of quadratic expressions:
- Multiply the factors (3x + 2) and (x - 4).
- Simplify the expression (x + 5)(x - 3).
- Compare the simplified expression with the options given and select the correct answer.
References
- "Algebra" by Michael Artin
- "Calculus" by Michael Spivak
- "Mathematics for the Nonmathematician" by Morris Kline
Glossary
- Quadratic expression: A polynomial of degree two, typically written in the form of ax² + bx + c.
- Distributive property: A property of multiplication that states a(b + c) = ab + ac.
- Like terms: Terms that have the same variable and exponent.
Understanding Quadratic Expressions
Quadratic expressions are a fundamental concept in algebra, and they play a crucial role in solving various mathematical problems. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It is typically written in the form of ax² + bx + c, where a, b, and c are constants, and x is the variable.
Frequently Asked Questions
Q: What is a quadratic expression?
A: A quadratic expression is a polynomial of degree two, typically written in the form of ax² + bx + c.
Q: How do I multiply quadratic expressions?
A: To multiply quadratic expressions, use the distributive property to break down the multiplication into smaller steps. Combine like terms to simplify the expression.
Q: What is the distributive property?
A: The distributive property is a property of multiplication that states a(b + c) = ab + ac.
Q: How do I simplify a quadratic expression?
A: To simplify a quadratic expression, combine like terms. Like terms are terms that have the same variable and exponent.
Q: What is the difference between a quadratic expression and a linear expression?
A: A quadratic expression is a polynomial of degree two, while a linear expression is a polynomial of degree one.
Q: Can you give an example of a quadratic expression?
A: Yes, an example of a quadratic expression is 2x² + 3x - 4.
Q: How do I factor a quadratic expression?
A: To factor a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Q: What is the vertex form of a quadratic expression?
A: The vertex form of a quadratic expression is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.
Q: How do I find the vertex of a quadratic expression?
A: To find the vertex of a quadratic expression, use the formula x = -b / 2a.
Q: Can you give an example of a quadratic expression in vertex form?
A: Yes, an example of a quadratic expression in vertex form is y = 2(x - 1)² + 3.
Q: How do I graph a quadratic expression?
A: To graph a quadratic expression, use the vertex form and plot the vertex. Then, plot two points on either side of the vertex and draw a smooth curve through them.
Q: What is the axis of symmetry of a quadratic expression?
A: The axis of symmetry of a quadratic expression is a vertical line that passes through the vertex of the parabola.
Q: Can you give an example of a quadratic expression with an axis of symmetry?
A: Yes, an example of a quadratic expression with an axis of symmetry is y = 2(x - 2)² + 1.
Conclusion
In this article, we have answered some of the most frequently asked questions about quadratic expressions. We have covered topics such as multiplying quadratic expressions, simplifying quadratic expressions, factoring quadratic expressions, and graphing quadratic expressions. We hope that this article has been helpful in clarifying any confusion you may have had about quadratic expressions.
Tips and Tricks
- When multiplying quadratic expressions, use the distributive property to break down the multiplication into smaller steps.
- Combine like terms to simplify the expression.
- Use the vertex form to graph a quadratic expression.
- Find the axis of symmetry to determine the vertex of a quadratic expression.
Real-World Applications
Quadratic expressions have numerous real-world applications, including:
- Modeling the motion of objects under the influence of gravity
- Describing the shape of a parabola
- Solving optimization problems in economics and business
Practice Problems
Try solving the following practice problems to reinforce your understanding of quadratic expressions:
- Multiply the factors (3x + 2) and (x - 4).
- Simplify the expression (x + 5)(x - 3).
- Compare the simplified expression with the options given and select the correct answer.
References
- "Algebra" by Michael Artin
- "Calculus" by Michael Spivak
- "Mathematics for the Nonmathematician" by Morris Kline
Glossary
- Quadratic expression: A polynomial of degree two, typically written in the form of ax² + bx + c.
- Distributive property: A property of multiplication that states a(b + c) = ab + ac.
- Like terms: Terms that have the same variable and exponent.
- Vertex form: A form of a quadratic expression that is written as y = a(x - h)² + k, where (h, k) is the vertex of the parabola.
- Axis of symmetry: A vertical line that passes through the vertex of a parabola.