Math Challenge: Solve Equations & Simplify Expressions!

Hey math enthusiasts! Ready to put your skills to the test? This article presents a fun and challenging set of math problems covering equation solving and expression simplification. Let's dive in and conquer these problems together!

P1: Solving the Equation (3x+7)/2 - 9 = (4x+1)/3

In this first problem, our main goal is to solve the equation (3x+7)/2 - 9 = (4x+1)/3. To kick things off, we need to get rid of those pesky fractions. The best way to do this is by finding the least common multiple (LCM) of the denominators, which are 2 and 3. The LCM of 2 and 3 is 6, so we'll multiply both sides of the equation by 6. Doing so helps us eliminate the fractions and simplifies the equation, making it much easier to handle. This is a crucial first step in solving any equation involving fractions, as it transforms the equation into a more manageable form. Remember, the goal is always to isolate the variable, in this case, x, so we need to strategically manipulate the equation using mathematical operations. By multiplying by the LCM, we set the stage for a smoother solving process. After multiplying both sides by 6, we get a new equation without fractions, which is a significant step forward. Next, we distribute the 6 on both sides of the equation. This means multiplying 6 by each term inside the parentheses. Distributing correctly is super important to maintain the balance of the equation and ensure we're on the right track to finding the correct solution. Mistakes in distribution can lead to an incorrect final answer, so double-check your work! The result of this distribution gives us a clearer picture of the equation's structure and allows us to proceed with combining like terms. This step simplifies the equation further, bringing us closer to isolating x. Combining like terms involves grouping terms with the same variable (x) and constant terms separately. This process helps to tidy up the equation and makes it easier to see the next steps required to solve for x. Think of it like organizing your workspace before tackling a complex task – a clean and organized equation is much easier to solve! After combining like terms, we move all the x terms to one side of the equation and the constant terms to the other side. This is a fundamental technique in solving equations, as it helps to isolate the variable we're trying to find. To do this, we'll use addition or subtraction to move terms across the equals sign. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain balance. This ensures that the equation remains valid and the solution we find is accurate. Once we've grouped the x terms and constants, we simplify both sides of the equation. This might involve adding, subtracting, or combining numbers to get a single term on each side. Simplification is key to making the equation as straightforward as possible, which helps in the final step of solving for x. A simplified equation reduces the chances of making errors and makes the solution more apparent. Finally, to isolate x, we divide both sides of the equation by the coefficient of x. The coefficient is the number that's multiplying x. Dividing by the coefficient effectively undoes the multiplication and leaves x by itself on one side of the equation. This step gives us the value of x that satisfies the original equation. Make sure to perform the division carefully and double-check your answer to ensure it's correct! And that's it – we've successfully solved for x!

P2: Expanding the Expression (a+b)^3

Now, let's tackle the expansion of the expression (a+b)^3. This problem involves applying the binomial theorem or, more simply, multiplying the expression (a+b) by itself three times. We can break this down into manageable steps to avoid errors. First, let's consider what (a+b)^3 actually means. It's the same as (a+b) * (a+b) * (a+b). We'll start by multiplying the first two (a+b) terms together. This is a classic example of using the distributive property, also known as the FOIL method (First, Outer, Inner, Last), which ensures that each term in the first binomial is multiplied by each term in the second binomial. Mastering this technique is essential for expanding binomial expressions correctly. When we multiply (a+b) by (a+b), we get a^2 + 2ab + b^2. Make sure to double-check your multiplication and combine like terms to arrive at this result. This intermediate step simplifies the overall expansion and prepares us for the next multiplication. The next step is to multiply the result (a^2 + 2ab + b^2) by the remaining (a+b) term. This is where the expansion gets a bit more involved, but sticking to the distributive property will guide us through. We need to ensure that each term in the trinomial (a^2 + 2ab + b^2) is multiplied by each term in the binomial (a+b). This process requires careful attention to detail to avoid any mistakes. Multiplying each term systematically is the key to success here. After performing the multiplication, we will have several terms. Our next task is to identify and combine the like terms. Like terms are terms that have the same variables raised to the same powers. Combining them simplifies the expression and brings us closer to the final expanded form. This is a crucial step in ensuring our final answer is in its simplest form. By carefully combining like terms, we avoid unnecessary complexity and make the expression more readable. The final expanded form of (a+b)^3 is a^3 + 3a^2b + 3ab^2 + b^3. This is a standard algebraic identity that is very useful to remember. Knowing this expansion by heart can save time in future problems and is a valuable tool in your mathematical toolkit. Understanding how to derive it, however, is even more important, as it reinforces your understanding of algebraic manipulation. So, there you have it – the expansion of (a+b)^3!

P3: Expanding the Expression (sqrt(2) - sqrt(3))^3

Moving on, let's expand the expression (sqrt(2) - sqrt(3))^3. This problem is similar to the previous one, but it involves square roots, which adds a little twist. We'll use the same strategy of multiplying the expression by itself three times, but we need to be careful when dealing with the square roots. Just like before, (sqrt(2) - sqrt(3))^3 means (sqrt(2) - sqrt(3)) * (sqrt(2) - sqrt(3)) * (sqrt(2) - sqrt(3)). We'll start by multiplying the first two (sqrt(2) - sqrt(3)) terms. This involves using the distributive property (FOIL method) again, but this time with square roots. Remember the rules for multiplying square roots: sqrt(a) * sqrt(b) = sqrt(ab). Applying this rule correctly is essential to getting the right result. When we multiply (sqrt(2) - sqrt(3)) by (sqrt(2) - sqrt(3)), we get 2 - 2sqrt(6) + 3, which simplifies to 5 - 2sqrt(6). Make sure you're comfortable with multiplying and simplifying square roots; it's a fundamental skill in algebra. This intermediate result prepares us for the next multiplication step. Now, we need to multiply (5 - 2sqrt(6)) by the remaining (sqrt(2) - sqrt(3)) term. This is where things get a bit more challenging, as we need to distribute carefully and combine terms involving square roots. Taking it one step at a time and being methodical will help prevent errors. We'll distribute each term in the first expression to each term in the second expression. After distributing, we'll have several terms, some with square roots and some without. Our next step is to simplify and combine like terms. This is crucial for arriving at the final simplified expression. Remember that we can only combine terms that have the same square root part. For example, we can combine 2sqrt(2) and 3sqrt(2), but we can't combine 2sqrt(2) with 3sqrt(3). Combining like terms correctly is key to simplifying the expression as much as possible. After simplifying and combining like terms, we arrive at the final expanded form: 11sqrt(2) - 9sqrt(3). This is the simplified version of (sqrt(2) - sqrt(3))^3. And there we have it! We've successfully expanded and simplified an expression involving square roots.

P4: Simplifying the Expression (5+2 sqrt(3))/sqrt(3) + (7-sqrt(2))/sqrt(2)

Lastly, let's simplify the expression (5+2 sqrt(3))/sqrt(3) + (7-sqrt(2))/sqrt(2). This problem involves rationalizing the denominator, which means getting rid of the square roots in the denominator of each fraction. To rationalize the denominator, we multiply both the numerator and the denominator of each fraction by the square root in the denominator. This is a standard technique for simplifying expressions with square roots in the denominator. By multiplying by the appropriate square root, we create a perfect square in the denominator, effectively eliminating the square root. For the first fraction, (5+2 sqrt(3))/sqrt(3), we multiply both the numerator and the denominator by sqrt(3). This gives us (5sqrt(3) + 23)/3, which simplifies to (5sqrt(3) + 6)/3. Rationalizing the denominator here makes the fraction easier to work with. For the second fraction, (7-sqrt(2))/sqrt(2), we multiply both the numerator and the denominator by sqrt(2). This gives us (7sqrt(2) - 2)/2. Again, rationalizing the denominator simplifies the fraction and prepares us for the next step. Now that we've rationalized both denominators, we have (5sqrt(3) + 6)/3 + (7sqrt(2) - 2)/2. To add these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6, so we'll rewrite each fraction with a denominator of 6. Finding a common denominator is essential for adding or subtracting fractions. It ensures that we're adding like parts, just like we need common units when adding measurements. To rewrite the first fraction with a denominator of 6, we multiply both the numerator and the denominator by 2. To rewrite the second fraction with a denominator of 6, we multiply both the numerator and the denominator by 3. This gives us a common denominator and allows us to combine the fractions. After rewriting the fractions with a common denominator, we can add the numerators. This involves distributing any multiplication and combining like terms. Remember to keep the denominator the same when adding fractions. The result of adding the numerators will give us a single fraction with a denominator of 6. After adding the fractions and simplifying, we get the final simplified expression: (10sqrt(3) + 21sqrt(2) + 6)/6. This is the simplified form of the original expression. Great job! We've successfully simplified an expression involving rationalizing denominators and adding fractions.

Conclusion

So, guys, we've conquered some challenging math problems today! We solved an equation, expanded binomial expressions with and without square roots, and simplified an expression by rationalizing the denominator. Keep practicing these skills, and you'll become a math whiz in no time! Remember, math is like a muscle – the more you use it, the stronger it gets. Keep challenging yourself, and you'll be amazed at what you can achieve!