☝️ Practice

☝️ Step-by-Step Guide

📝 The Problem

You're looking to find the value(s) of $x$ that satisfy the equation $x = x^2$. This is a classic algebraic equation that helps illustrate some fundamental problem-solving techniques.

☝️ The Plan

Here's a straightforward approach to tackle this:

1. Rearrange the Equation: Think about how to get all the terms onto one side of the equals sign, usually setting the equation to zero. This is a common first step for solving polynomial equations.
2. Factor: Once the equation is set to zero, consider if there are common factors you can pull out. Factoring is a powerful tool for finding solutions.
3. Solve for x: After factoring, you'll have expressions that, when multiplied, equal zero. Remember the zero product property: if a product of factors is zero, at least one of the factors must be zero.
4. Verify Solutions: It's always a good idea to plug your potential solutions back into the original equation to make sure they work.

🚀 Let's Get Started

Let's begin with that first step. Your goal is to rearrange the equation $x = x^2$ so that one side is zero.

Take a moment to think about how you might do that.

💬 Discussion

I'm happy to guide you, but I need to see your attempt first. Can you show me how you would rearrange $x = x^2$ to get zero on one side?

💬 Discussion

You divided by $x$ and got $1=x$. That's a neat trick for finding one of the answers, $x=1$.

But tell me, what happens if $x$ happens to be zero? Can we divide by zero?

That's why moving everything to one side, like $x^2 - x = 0$, is generally the more robust method. It helps us catch all the solutions. What does that $x^2 - x = 0$ form reveal that dividing by $x$ might have hidden?

💬 Discussion

No worries, we can work through it. We're looking at the equation $x^2 - x = 0$.

Can you identify anything common between the terms $x^2$ and $-x$? What might you be able to pull out from both of them?

💬 Discussion

Ah, you've zeroed in on $x$! That's exactly right.

So, if $x$ is the common factor in $x^2 - x$, what does that expression look like when you factor out the $x$?

Think about it like this: $x^2$ is $x \cdot x$, and $-x$ is $-1 \cdot x$. What's left over when you pull the $x$ out from both terms?

💬 Discussion

You've nailed it! Factoring $x^2 - x$ indeed gives you $x(x-1)$. Nicely done.

So now we have the equation in its factored form:

$$x(x-1) = 0$$

What does this tell us about the values $x$ can take? Remember what happens when you multiply things together and the result is zero.

💬 Discussion

🥳 You've got it exactly right!

You've successfully navigated the steps to find the solutions. Here's a quick recap of your journey:

1. You started with the equation $x = x^2$
2. You rearranged it to get all terms on one side, forming $x^2 - x = 0$
3. You factored out the common term $x$, leading to $x(x-1) = 0$
4. By applying the zero product property, you correctly deduced that for the product to be zero, either $x=0$ or $(x-1)=0$, giving you the solutions $x=0$ or $x=1$

Excellent work!