Math Mistake: Find The First Error In Solving Equations

Hey math whizzes and problem-solvers! Ever feel like you're on the verge of cracking a tough problem, only to realize you've taken a wrong turn somewhere? It happens to the best of us, especially when we're diving into algebra. Today, we're going to dissect a common type of math error that pops up when solving equations, specifically focusing on quadratic equations. Let's get our detective hats on and figure out exactly where a student might have first stumbled in the process of solving the equation $5 x^2+45 x=0$. This isn't just about finding the mistake; it's about understanding why it's a mistake, so we can all level up our equation-solving game. We'll break down each step, explain the math behind it, and pinpoint the exact moment things went off track. So, grab your calculators (or just your brilliant brains!), and let's dive into this mathematical mystery together. We'll be looking at the steps provided and determining if the error occurred in Step 1, Step 2, or Step 3. Get ready to boost your algebra skills, guys!

The Problem: Unpacking the Equation $5 x^2+45 x=0$

Alright team, let's start by looking at the original equation we're working with: $5 x^2+45 x=0$. This is a quadratic equation, which means it has a term with $x^2$. The goal here is to find the values of $x$ that make this equation true. Notice that the equation is already set equal to zero, which is a super helpful starting point for solving many types of quadratic equations. The presence of the $x^2$ term and the $x$ term, with no constant term (like a plain number without an $x$), suggests that factoring is likely going to be a key strategy. This form of the equation is what we call standard form ($ax^2 + bx + c = 0$), where in this case, $a=5$, $b=45$, and $c=0$. When $c=0$, it often means that $x=0$ is one of the solutions, and factoring out a common factor involving $x$ is the way to go. So, before we even look at the steps the student took, we know the equation is ripe for factoring. The common factors here are clearly a $5$ and an $x$. Recognizing this upfront is crucial because it sets the expectation for how the equation should be manipulated. We expect to see these common factors pulled out, leading to a product of terms that equals zero. This initial understanding helps us evaluate the subsequent steps more critically. We're not just blindly following along; we're applying our knowledge of algebraic principles to assess the validity of each transformation. So, keep this initial equation in mind as we move through the student's work. The goal is to find values of $x$ that satisfy this specific relationship between $x^2$ and $x$.

Step 1: Factoring Made Easy? ($5 x(x+9)=0$)

Now, let's look at the first step the student took: Step 1: $5 x(x+9)=0$. This step involves factoring the original expression $5 x^2+45 x$. To check if this step is correct, we need to see if the expression $5x(x+9)$ expands back to the original $5 x^2+45 x$. Let's use the distributive property (or think of it as FOIL if you prefer, though it's simpler here). We multiply the $5x$ outside the parentheses by each term inside the parentheses:

• $5x imes x = 5x^2$
• $5x imes 9 = 45x$

Combining these, we get $5x^2 + 45x$. Boom! This matches the original expression exactly. This means the student correctly factored the quadratic expression in Step 1. They identified the greatest common factor (GCF) of $5x^2$ and $45x$, which is indeed $5x$. When they factored out $5x$, they were left with $x$ from the $5x^2$ term (since $5x^2 / 5x = x$) and $9$ from the $45x$ term (since $45x / 5x = 9$). So, the factored form $5x(x+9)$ is accurate. This is a critical step because the Zero Product Property relies on having an equation in the form of (factor 1) * (factor 2) * ... = 0. If the factoring is wrong, the subsequent steps will almost certainly be incorrect, even if they appear mathematically sound on their own. The student successfully applied this factoring technique, demonstrating a solid understanding of how to simplify polynomial expressions by finding common factors. This makes Step 1 a success, and it means the error, if there is one, must lie further down the line. It's always good to double-check factoring, as it's a common place for slips, but in this case, the student nailed it. Keep that $5x(x+9)=0$ form handy as we move to the next stage of solving.

Step 2: The Crucial Split ($5 x=0$ and $x+9=0$)

Moving on, let's examine Step 2: $5 x=0$ and $x+9=0$. This step is where the Zero Product Property comes into play. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In Step 1, we had the product $5x imes (x+9)$ equaling zero. According to the Zero Product Property, this means either the first factor ($5x$) is zero, OR the second factor ($x+9$) is zero (or both, but that's covered by the 'or'). So, the student correctly set up two separate, simpler equations, one for each factor, both equal to zero. This is the standard and correct procedure when solving factored polynomial equations. The logic here is sound: if $A imes B = 0$, then $A=0$ or $B=0$. Here, $A = 5x$ and $B = (x+9)$. Therefore, setting $5x = 0$ and $x+9 = 0$ is the mathematically correct application of the Zero Product Property. This step transforms the single, more complex quadratic equation into two independent linear equations, each of which is much easier to solve. The student has accurately identified that both possibilities must be considered to find all possible solutions for $x$. This is a pivotal moment in solving equations of this type, and the student has navigated it perfectly. There's no error here; this is exactly what needs to be done. The accuracy of this step reinforces that the foundation is solid, and we should continue to the final step to see if the solutions derived from these linear equations are correct.

Step 3: Finding the Solutions ($x=0$ and $x=9$)

Finally, let's scrutinize Step 3: $x=0$ and $x=9$. This step involves solving the two linear equations that were set up in Step 2: $5x=0$ and $x+9=0$. Let's tackle them one by one.

• First equation: $5x = 0$ To solve for $x$, we need to isolate it. We can do this by dividing both sides of the equation by 5. $5x / 5 = 0 / 5$ $x = 0$ This part of Step 3 is correct. The student correctly found that $x=0$ is one solution.

• Second equation: $x+9 = 0$ To solve for $x$, we need to isolate it. We can do this by subtracting 9 from both sides of the equation. $x + 9 - 9 = 0 - 9$ $x = -9$

Now, compare this result to what the student wrote in Step 3. The student wrote $x=0$ and $x=9$. We found $x=0$ is correct. However, for the second equation, $x+9=0$, the correct solution is $x = -9$, not $x=9$. The student seems to have incorrectly concluded that $x+9=0$ implies $x=9$. This is a common mistake where students forget to carry the negative sign when isolating the variable, or they mistakenly change the sign of the constant term when moving it to the other side of the equation. The correct operation is subtracting 9 from both sides, which results in $x = -9$. Therefore, the error occurs in Step 3 when solving the second linear equation.

Conclusion: Where Did the Student First Go Wrong?

We've carefully analyzed each step. In Step 1, the student correctly factored the quadratic expression $5x^2+45x$ into $5x(x+9)$. This step was executed perfectly. In Step 2, the student correctly applied the Zero Product Property by splitting the factored equation into two separate linear equations: $5x=0$ and $x+9=0$. This is the standard and appropriate method for solving factored equations. The mistake was found in Step 3, where the student attempted to solve these two linear equations. While the solution for $5x=0$ yielded the correct result of $x=0$, the solution for $x+9=0$ was incorrect. Instead of obtaining $x=-9$ (by subtracting 9 from both sides), the student mistakenly concluded that $x=9$. This is the first and only error in the sequence provided. The student's final answer should have been $x=0$ and $x=-9$, not $x=0$ and $x=9$. Therefore, the student first made an error in C. Step 3.

Keep practicing, guys! Recognizing these common errors is a huge step in mastering algebra. Happy solving!