Solving Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's break down these math problems together. We're going to tackle some algebraic expressions and solve for x where needed. Don't worry; it's easier than it looks! We'll cover each expression step by step, so grab your pencils, and let's dive in!

a) (2a)(-3a)

Let's start with our first expression: (2a)(-3a). This looks like a simple multiplication problem involving variables. The key here is to multiply the coefficients (the numbers in front of the variables) and then handle the variables themselves. So, we have 2 multiplied by -3, which gives us -6. Then, we have a multiplied by a, which gives us a². Putting it all together, the simplified expression is -6a². Remember, when multiplying variables with the same base, you add their exponents. In this case, a has an exponent of 1, so a¹ * a¹ = a^(1+1) = a². This principle is fundamental in algebra and is used extensively in various mathematical contexts, including polynomial manipulations, calculus, and complex analysis. Think of it as combining like terms but in a multiplicative way. For example, if you had (4b²)(3b³), you'd multiply 4 and 3 to get 12, and then multiply b² and b³ to get b^(2+3) = b⁵. Thus, the simplified expression would be 12b⁵. Understanding this concept is crucial for simplifying more complex algebraic expressions and solving equations effectively. This basic rule extends to more complex scenarios involving multiple variables and exponents, forming the foundation for advanced algebraic techniques and problem-solving strategies. So, the final answer is indeed -6a². It is short and concise and accurately reflects the simplified form of the original expression. Remember, paying close attention to signs (positive or negative) and exponents is vital to avoid common errors in algebraic manipulations. Keep practicing, and you'll become a pro in no time!

b) (-0.3x+3)(5x-2)

Next up, we have (-0.3x + 3)(5x - 2). This is a binomial multiplication, which means we need to use the distributive property (often remembered as the FOIL method: First, Outer, Inner, Last). Let's break it down:

• First: Multiply the first terms in each binomial: (-0.3x) * (5x) = -1.5x²
• Outer: Multiply the outer terms: (-0.3x) * (-2) = 0.6x
• Inner: Multiply the inner terms: (3) * (5x) = 15x
• Last: Multiply the last terms: (3) * (-2) = -6

Now, combine all the terms: -1.5x² + 0.6x + 15x - 6. Combine the like terms (the x terms): 0.6x + 15x = 15.6x. So, the simplified expression is -1.5x² + 15.6x - 6. This method, often called the FOIL method, is a systematic way to ensure that each term in the first binomial is multiplied by each term in the second binomial. Understanding and mastering this technique is crucial for expanding more complex polynomial expressions and simplifying equations. For example, if you had (2y + 1)(3y - 4), you would multiply 2y by 3y to get 6y², 2y by -4 to get -8y, 1 by 3y to get 3y, and 1 by -4 to get -4. Combining these terms, you would get 6y² - 8y + 3y - 4, which simplifies to 6y² - 5y - 4. This approach is also applicable to expressions involving more than two terms, where the distributive property is applied iteratively to expand the expression completely. Practice is key to becoming proficient in using the distributive property and simplifying polynomial expressions accurately. This skill is essential for various mathematical applications, including solving quadratic equations, graphing functions, and performing calculus operations. The expression -1.5x² + 15.6x - 6 represents a quadratic expression, which can be further analyzed to find its roots, vertex, and other key features. Remember to always double-check your work and ensure that you have combined all like terms correctly. Keep practicing, and you'll become more confident in expanding and simplifying algebraic expressions.

c) (-√7m)(√7m)

Okay, let's tackle (-√7m)(√7m). This involves square roots, but don't let that scare you! Remember that √a * √a = a. In this case, we have -√7 multiplied by √7, which equals -7. Then we have m multiplied by m, which equals m². Combining these, the simplified expression is -7m². When dealing with square roots, it's important to remember that the square root of a number multiplied by itself gives you the original number. This is because the square root operation is the inverse of squaring a number. For example, √5 * √5 = 5. In this expression, the presence of the negative sign in front of the first term is crucial, as it affects the sign of the final result. This principle is widely used in simplifying expressions involving radicals and rationalizing denominators. Additionally, understanding how to manipulate square roots is essential for solving equations involving radicals and working with irrational numbers. This expression provides a good example of how square roots and variables can be combined in algebraic expressions. By applying the rules of algebra and the properties of square roots, we can easily simplify the expression to its final form. The expression -7m² represents a quadratic term with a negative coefficient, indicating that the parabola opens downward. This can be important in applications such as finding the maximum value of a function or modeling physical phenomena. Remember to always pay attention to the signs and exponents when working with algebraic expressions. Keep practicing, and you'll become more confident in simplifying expressions involving square roots and variables.

d) 5 |(2.5x)|

Lastly, let's solve for x in 5|(2.5x)|. The vertical bars mean absolute value. This means the expression inside the bars, (2.5x), can be either positive or negative, but the result will always be non-negative. First, divide both sides by 5: |2.5x| = 1. Now, we have two cases to consider:

• Case 1: 2.5x = 1. Divide both sides by 2.5: x = 1 / 2.5 = 0.4
• Case 2: 2.5x = -1. Divide both sides by 2.5: x = -1 / 2.5 = -0.4

Therefore, the solutions are x = 0.4 and x = -0.4. When solving equations involving absolute values, it is essential to consider both the positive and negative cases. This is because the absolute value of a number is its distance from zero, which means that both the positive and negative values of the number will have the same absolute value. For example, |3| = 3 and |-3| = 3. In this equation, we need to find the values of x that make the expression inside the absolute value equal to both 1 and -1. By solving these two equations separately, we can find all possible solutions for x. The solutions x = 0.4 and x = -0.4 represent the points where the expression inside the absolute value is equal to 1 or -1. These solutions are important in various applications, such as finding the zeros of a function or modeling physical phenomena. Remember to always check your solutions by plugging them back into the original equation to ensure that they satisfy the equation. Keep practicing, and you'll become more confident in solving equations involving absolute values. The concept of absolute value is widely used in mathematics, physics, and engineering, and it is important to have a solid understanding of how to work with it. In addition, solving equations with absolute values can be extended to more complex scenarios involving inequalities and systems of equations.

So, there you have it! We've solved each expression step by step. Keep practicing these kinds of problems, and you'll become a math whiz in no time! Good luck, and have fun solving!