Marcus's Quadratic Equation Solution: Spot The Error!

by Andrew McMorgan 54 views

Hey Plastik Magazine readers! Let's dive into a quadratic equation problem today. We're going to analyze a student's attempt to solve a quadratic equation and see if we can spot any mistakes. It's like being a math detective, and it's super fun! So, buckle up, and let's get started!

The Problem: Marcus's Quadratic Journey

So, our friend Marcus was tackling the quadratic equation x² - 10x + 25 = 0. Quadratic equations, for those who need a quick refresher, are equations of the form ax² + bx + c = 0, where a, b, and c are constants. The solutions to these equations are the values of x that make the equation true. Marcus decided to use the quadratic formula to solve this, which is a perfectly valid approach. The quadratic formula is like the Swiss Army knife of quadratic equations – it works every time! It's given by:

x = (-b ± √(b² - 4ac)) / 2a

Marcus's steps were as follows:

  1. x = (-b ± √(b² - 4ac)) / 2a
  2. x = (-(10) ± √((10)² + 4(1)(25))) / 2(1)
  3. x = (-10 ± √200) / 2

Now, at first glance, this might seem okay. But remember, in math, the devil is in the details! We need to carefully examine each step to make sure everything is correct. Did Marcus make a mistake? That's our mission to find out!

Step-by-Step Breakdown: Spotting the Flaw

Alright, let's put on our detective hats and break down Marcus's solution step-by-step. We need to scrutinize every move to see where things might have gone sideways. Remember, even a small error early on can throw off the entire solution, so we've got to be meticulous.

Step 1: The Quadratic Formula

The first step, x = (-b ± √(b² - 4ac)) / 2a, is simply stating the quadratic formula. This is the correct formula, so no problems here. Marcus knows his formula, which is a great start! It's like having the right map before you begin your treasure hunt. So far, so good.

Step 2: Substitution - Uh Oh, Houston, We Have a Problem!

Now, this is where things get interesting. In the equation x² - 10x + 25 = 0, we can identify a = 1, b = -10, and c = 25. The quadratic formula requires us to substitute these values correctly. Let's see what Marcus did:

x = (-(10) ± √((10)² + 4(1)(25))) / 2(1)

Do you see it? The problem lies within the square root. Marcus has substituted 'b' as 10 inside the square root, but the actual value of 'b' is -10. Remember, the formula has b², which means we need to square the entire value of b, including the negative sign if there is one. This is a classic mistake, and it's super important to catch it!

The correct substitution should have been:

x = (-(-10) ± √((-10)² - 4(1)(25))) / 2(1)

Notice the difference? We've got a -(-10) outside the square root and a (-10)² inside. These little changes make a big difference in the final answer.

Step 3: Simplification - Following the Ripple Effect

Marcus's third step, x = (-10 ± √200) / 2, is a direct consequence of the incorrect substitution in Step 2. Since the values inside the square root were wrong to begin with, the result is also incorrect. The square root of 200 comes from the wrong calculation inside the square root in the previous step. This shows how one small mistake can create a ripple effect, leading to further errors.

The Correct Solution: Let's Do It Right!

Okay, now that we've pinpointed Marcus's mistake, let's solve the equation correctly. This will help solidify our understanding and show how important it is to pay attention to detail.

  1. Correct Substitution:

    x = (-(-10) ± √((-10)² - 4(1)(25))) / 2(1)

  2. Simplify Inside the Square Root:

    x = (10 ± √(100 - 100)) / 2

  3. Further Simplification:

    x = (10 ± √0) / 2

  4. Final Solution:

    x = 10 / 2 = 5

So, the correct solution is x = 5. Notice that there's only one solution because the discriminant (the part inside the square root) is zero. This means the quadratic equation has a repeated root.

Key Takeaways: Lessons from Marcus's Mishap

So, what can we learn from Marcus's adventure with quadratic equations? A few key things:

1. Pay Attention to Signs

This is super crucial in math. A simple sign error can throw off the entire calculation. Always double-check your signs, especially when dealing with negative numbers.

2. Correct Substitution is Key

Make sure you're substituting the correct values into the formula. Double-check which value corresponds to a, b, and c in your equation.

3. Follow the Order of Operations

Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Doing operations in the wrong order can lead to errors.

4. Double-Check Your Work

It might seem tedious, but taking a few extra minutes to review your steps can save you from making silly mistakes. It's like proofreading an essay – you're bound to catch something you missed the first time around.

Conclusion: Math Detective Skills Activated!

Great job, guys! We successfully analyzed Marcus's work, identified the error, and solved the quadratic equation correctly. This exercise highlights the importance of precision and attention to detail in math. Remember, even experienced mathematicians make mistakes sometimes – the key is to learn from them and develop good problem-solving habits. Keep practicing, keep questioning, and most importantly, keep having fun with math! Until next time, keep those math detective skills sharp!