Solving For An Unknown Number Three Times The Square Decreased By Nine Times Equals 120
Hey there, math enthusiasts! Ever stumbled upon a word problem that seems like a riddle wrapped in an enigma? Well, today, we're diving headfirst into one such mathematical puzzle. We're going to dissect it, solve it, and most importantly, understand the logic behind it. So, grab your thinking caps, and let's embark on this mathematical journey together!
Decoding the Mathematical Enigma
The problem statement presents us with a fascinating scenario: "Three times the square of a certain number is decreased by 9 times the number. The result is 120. Find the number." At first glance, it might seem like a jumble of words and numbers, but don't worry, we'll break it down step by step.
Translating Words into Math
The first step in solving any word problem is to translate the given information into mathematical expressions. This is where our algebraic prowess comes into play. Let's represent the "certain number" we're trying to find with the variable x. Now, let's analyze the problem statement piece by piece:
- "Three times the square of a certain number": This translates to 3 * x^2, or simply 3x^2.
- "Decreased by": This indicates subtraction, so we'll use the minus sign (-).
- "9 times the number": This translates to 9 * x, or 9x.
- "The result is 120": This means the entire expression equals 120.
Putting it all together, we get the following equation:
3x^2 - 9x = 120
This equation is the heart of our problem. It encapsulates all the information given in the word problem in a concise mathematical form. Now that we have our equation, it's time to solve it.
Cracking the Quadratic Code
Notice that our equation, 3x^2 - 9x = 120, is a quadratic equation. Quadratic equations are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. To solve a quadratic equation, we first need to bring it into the standard form by setting it equal to zero. Let's do that:
3x^2 - 9x - 120 = 0
Now we can solve this quadratic equation using several methods, such as factoring, completing the square, or using the quadratic formula. For this particular equation, factoring is the most efficient approach. Let's see how it works.
Factoring for the Win
Before we jump into factoring, let's simplify the equation by dividing both sides by the greatest common factor, which is 3:
x^2 - 3x - 40 = 0
Now, we need to find two numbers that multiply to -40 and add up to -3. After a bit of thought, we can identify those numbers as -8 and 5.
So, we can factor the quadratic expression as follows:
(x - 8)(x + 5) = 0
For the product of two factors to be zero, at least one of them must be zero. This gives us two possible solutions:
- x - 8 = 0, which means x = 8
- x + 5 = 0, which means x = -5
Therefore, the solutions to our quadratic equation are x = 8 and x = -5. But what do these solutions mean in the context of our original word problem?
The Numbers Revealed
We've found two possible values for the "certain number" mentioned in the problem: 8 and -5. Both of these numbers satisfy the conditions stated in the problem.
Let's check if these solutions work:
- For x = 8: 3(8^2) - 9(8) = 3(64) - 72 = 192 - 72 = 120. Bingo!
- For x = -5: 3((-5)^2) - 9(-5) = 3(25) + 45 = 75 + 45 = 120. Another bullseye!
So, the numbers we were looking for are indeed 8 and -5. We've successfully cracked the mathematical enigma!
Mastering the Art of Word Problems
Word problems might seem daunting at first, but with a systematic approach and a bit of practice, they become much more manageable. Here are some key strategies to keep in mind when tackling word problems:
- Read Carefully: The first and most crucial step is to read the problem statement carefully and understand what it's asking. Identify the key information and what you need to find.
- Translate into Math: Convert the words into mathematical expressions and equations. This is where you'll use variables to represent unknown quantities and mathematical symbols to represent operations.
- Solve the Equation: Once you have an equation, use your algebraic skills to solve for the unknown variable(s). Remember to use the appropriate methods based on the type of equation you're dealing with.
- Check Your Answer: After you've found a solution, it's essential to check if it makes sense in the context of the original problem. Plug your solution back into the equation and see if it satisfies the conditions.
- Practice Makes Perfect: The more word problems you solve, the better you'll become at understanding and tackling them. Don't be afraid to ask for help when you're stuck, and keep practicing!
Real-World Math Adventures
Math isn't just about numbers and equations; it's a powerful tool that helps us understand and solve problems in the real world. Word problems are a great way to see how math concepts apply to everyday situations.
Think about it: When you're calculating the tip at a restaurant, figuring out the discount on a sale item, or planning a road trip, you're using math! Word problems help you develop critical thinking skills and the ability to apply mathematical knowledge to practical scenarios.
So, the next time you encounter a word problem, don't shy away from it. Embrace the challenge, break it down, and enjoy the satisfaction of finding the solution!
Wrapping Up: The Power of Mathematical Thinking
Solving the problem of finding the number that satisfies the given conditions was more than just a mathematical exercise. It was an exercise in critical thinking, problem-solving, and logical reasoning. These skills are invaluable not only in mathematics but in all aspects of life.
By translating the words into equations, factoring the quadratic expression, and interpreting the solutions, we not only found the answer but also deepened our understanding of mathematical concepts. And that's the true power of mathematical thinking – the ability to unravel complex problems and arrive at elegant solutions.
So, keep exploring the world of mathematics, keep challenging yourself with new problems, and never stop learning. Who knows what mathematical adventures await you!
