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In mathematics, a rational expression is considered 'undefined' when its denominator is zero. This is because division by zero is not possible, leading to an undefined result. For example, in the expression \(\frac{x^2}{4x-12}\), the expression will be undefined if the denominator \(4x - 12\) equals zero. We can solve for when this happens by setting \(4x - 12\) equal to zero. Let's break this down further:

\[4x - 12 = 0\]
In this case, solving for \(x\) will give us the value that makes the expression undefined. Adding 12 to both sides of the equation, we get \[4x = 12\]. When we divide both sides by 4, we find that \[x = 3\].

Therefore, the expression \(\frac{x^2}{4x-12}\) is undefined when \(x = 3\). Remember, the key to determining undefined expressions is always to check where the denominator equals zero!

The denominator in a rational expression is the term located at the bottom part of the fraction. For example, in the rational expression \(\frac{x^2}{4x-12}\), the denominator is \(4x - 12\). The denominator is crucial because it tells us how the value changes as the variable in the expression changes.

To find out if a rational expression is undefined, you need to determine when the denominator is equal to zero. In the given example, setting \(4x - 12\) to zero and solving for \(x\) allows us to find the value that makes the expression undefined. This process often involves simple algebraic steps:

• Set the denominator equal to zero: \[4x - 12 = 0\]
• Solve for \(x\) by isolating the variable: \[4x = 12\], then \[x = 3\].

Once you have done this, you know that whenever \(x = 3\), the denominator will be zero, rendering the entire expression undefined.

Solving equations is a fundamental skill in mathematics, often used to find values that make expressions true or false. In our context, we used equation-solving to identify when the denominator of a rational expression becomes zero.

Let's revisit the rational expression \(\frac{x^2}{4x-12}\):
We need to solve the equation involving the denominator to find when the expression is undefined:

• First, identify the denominator: \(4x - 12\).
• Set it equal to zero: \[4x - 12 = 0\]
• Solve for \(x\): Add 12 to both sides, resulting in \[4x = 12\]
• Divide by 4: \[x = 3\]

This method shows that \(x = 3\) makes the expression undefined because it sets the denominator to zero. These steps help simplify a possibly complex problem into manageable parts, aiding in understanding the broader concept.

Remember, solving these equations accurately involves carefully performing each algebraic operation. Whether we are solving for an undefined expression or another variable, mastering these steps is key to success in mathematics!