Is X = -3 A Solution To -3x^2 - 9x = 0?

When we're diving into the world of algebra, a common task is to figure out if a specific value, known as a root or a solution, actually makes an equation true. Today, we're tackling a quadratic equation: $-3x^2 - 9x = 0$. Our mission is to determine if $x=-3$ is a valid solution. This might seem like a simple plug-and-chug problem, but understanding why it is or isn't a solution is key to mastering quadratic equations. So, let's break down the process step-by-step, ensuring we understand the reasoning behind the answer. We'll be substituting the given value of $x$ into the equation and seeing if the left side magically equals the right side. If it does, hooray, we've found ourselves a solution! If not, well, that's okay too; it just means that particular number isn't the one we're looking for in this case. The beauty of mathematics is in its certainty, and checking solutions is a fundamental part of that certainty. We’ll go through this specific problem with $x=-3$ and see what happens, making sure to be meticulous with our calculations. This process is crucial for building a strong foundation in algebra, as it applies to countless other problems you'll encounter.

The Core Concept: Verifying Solutions

The fundamental principle we're using here is solution verification. In mathematics, a solution to an equation is a value (or set of values) for the variable that makes the equation a true statement. For our quadratic equation, $-3x^2 - 9x = 0$, we need to see if substituting $x=-3$ results in a true statement. A true statement in this context means that the expression on the left side of the equals sign will be exactly equal to the expression on the right side. The right side is a simple $0$, so our goal is to make the left side, $-3x^2 - 9x$, also equal to $0$ when $x=-3$. This verification process is not just about getting the right answer; it’s about understanding the logic behind it. It’s like a detective checking if a suspect was at the scene of the crime by looking at alibis and evidence. In math, our evidence is the equation, and our alibi is the proposed solution. The steps involved are straightforward: identify the equation, identify the proposed solution, substitute the solution into the equation, perform the calculations carefully, and finally, compare the results on both sides of the equals sign. If both sides match, the proposed value is indeed a solution. If they don't match, it's not. It's a rigorous but ultimately satisfying way to confirm our findings and build confidence in our mathematical abilities. This method is universal and applies to linear equations, quadratic equations, and indeed, almost any type of equation you might encounter in your mathematical journey. It's a cornerstone skill that will serve you well.

Step-by-Step Verification for $x=-3$

Let's roll up our sleeves and perform the substitution for the quadratic equation $-3x^2 - 9x = 0$ with our proposed solution $x=-3$. The first step is to carefully replace every instance of $x$ in the equation with the value $-3$. It's crucial to use parentheses when substituting, especially when dealing with negative numbers and exponents, to avoid sign errors. So, our equation becomes: $-3(-3)^2 - 9(-3) = 0$. Now, we follow the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). First, we handle the exponent: $(-3)^2$. Squaring a negative number always results in a positive number. So, $(-3)^2 = (-3) imes (-3) = 9$. Now, substitute this back into our equation: $-3(9) - 9(-3) = 0$. Next, we perform the multiplications. We have two multiplication operations: $-3(9)$ and $-9(-3)$. First, $-3 imes 9 = -27$. Then, $-9 imes (-3) = +27$. Now, substitute these results back into the equation: $-27 + 27 = 0$. Finally, we perform the addition: $-27 + 27$. This sum equals $0$. So, the equation becomes $0 = 0$. Since the left side of the equation ($0$) is indeed equal to the right side of the equation ($0$), the statement $0=0$ is true. This confirms that our initial proposed value, $x=-3$, is a valid solution to the quadratic equation $-3x^2 - 9x = 0$. This careful, step-by-step approach minimizes the chance of errors and clearly demonstrates why $x=-3$ works.

Analyzing the Reasoning: Why $x=-3$ Works (or Doesn't)

Now that we've gone through the calculation, let's articulate the reasoning clearly. Ariel's task is to determine if $x=-3$ is a solution. The core principle, as we've discussed, is substitution and verification. When Ariel substitutes $x=-3$ into the equation $-3x^2 - 9x = 0$, the following occurs:

1. Substitution: Replace $x$ with $-3$. This yields $-3(-3)^2 - 9(-3)$.
2. Exponent Calculation: Calculate $(-3)^2$. Since $(-3) imes (-3) = 9$, the expression becomes $-3(9) - 9(-3)$.
3. Multiplication: Perform the multiplications: $-3 imes 9 = -27$ and $-9 imes (-3) = +27$. The expression is now $-27 + 27$.
4. Addition: Calculate $-27 + 27$, which equals $0$.

Therefore, the original equation $-3x^2 - 9x = 0$ transforms into $0 = 0$ when $x=-3$. Because $0=0$ is a true statement, $x=-3$ is a solution to the equation. The reasoning that demonstrates this correctness involves showing these precise steps and concluding that the equality holds. If, after these steps, the left side did not equal the right side (e.g., if we got $5=0$), then $x=-3$ would not be a solution. The explanation must articulate that substituting the value leads to a mathematically correct identity (like $0=0$). Any explanation that correctly follows these substitution and calculation steps and arrives at a true statement is demonstrating the correct reasoning.

Alternative Approaches and Common Pitfalls

While direct substitution is the most straightforward way to verify a solution, it's worth considering how else one might approach this or what common mistakes could arise. For instance, one could first solve the quadratic equation $-3x^2 - 9x = 0$ for its roots and then check if $-3$ is among them. To solve it, we can factor out a common term, which is $-3x$: $-3x(x + 3) = 0$. For this product to be zero, at least one of the factors must be zero. So, we set each factor equal to zero:

• $-3x = 0 ightarrow x = 0$
• $x + 3 = 0 ightarrow x = -3$

By solving the equation first, we find that the solutions are $x=0$ and $x=-3$. This method confirms that $x=-3$ is indeed a solution. However, the question specifically asks for the explanation demonstrating correct reasoning for checking if $x=-3$ is a solution. This implies the verification process rather than the solving process itself. Common pitfalls in substitution include:

• Sign Errors: Forgetting that squaring a negative number yields a positive number (e.g., writing $(-3)^2 = -9$ instead of $9$).
• Order of Operations Errors: Performing operations in the wrong sequence.
• Incorrect Substitution: Mishandling the negative sign during multiplication (e.g., $-9(-3)$ becoming $-27$ instead of $+27$).

A correct explanation must show the accurate application of these rules. For example, simply stating "Yes, because $-3$ times $-3$ is $9$