Simplify the Rational Expression: (18x²-9x)/(2x-1)

Rational Expression Simplification with Factoring

Simplify the following expression:

18x29x2x1 \frac{18x^2-9x}{2x-1}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simply
00:03 Let's break down 18 into factors 9 and 2
00:06 Let's break down the power into multiplications
00:18 Let's mark the common factors
00:32 Let's take out the common factors from the parentheses
00:47 Let's reduce what we can
00:55 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Simplify the following expression:

18x29x2x1 \frac{18x^2-9x}{2x-1}

2

Step-by-step solution

Let's simplify the given expression:

18x29x2x1 \frac{18x^2-9x}{2x-1} Remember that we can reduce complete expressions only when both the numerator and denominator are completely factored into multiplication expressions,

For this, we'll use factorization, identify that in the numerator we can factor out a common term, do this, then reduce the expressions possible in the resulting fraction:

18x29x2x118x(2x1)2x118x \frac{18x^2-9x}{2x-1} \\ \frac{18x(2x-1)}{2x-1} \\ \downarrow\\ \boxed{18x} Therefore, the correct answer is answer B.

3

Final Answer

9x 9x

Key Points to Remember

Essential concepts to master this topic
  • Rule: Factor numerator and denominator completely before canceling terms
  • Technique: Factor out common term: 18x² - 9x = 9x(2x - 1)
  • Check: Multiply simplified answer by denominator: 9x(2x - 1) = 18x² - 9x ✓

Common Mistakes

Avoid these frequent errors
  • Canceling terms without complete factorization
    Don't cancel individual terms like canceling 18x² with 2x = wrong result! This ignores proper algebraic rules and creates incorrect expressions. Always factor completely first, then cancel only common factors that appear in both numerator and denominator.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

\( \frac{7}{7\cdot8}=8 \)

FAQ

Everything you need to know about this question

Why can't I just cancel the x terms directly?

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You can only cancel complete factors, not individual terms! The numerator 18x² - 9x is a sum, not a product, so you must factor it first to see what can actually be canceled.

How do I know when an expression is completely factored?

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An expression is completely factored when it's written as a product of factors that cannot be factored further. Look for common factors to pull out first, like 9x from 18x² - 9x.

What if the denominator doesn't factor out completely?

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That's fine! In this problem, after factoring the numerator as 9x(2x1) 9x(2x-1) , the factor (2x-1) cancels with the denominator, leaving just 9x.

Can this expression be simplified further than 9x?

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No! 9x is the completely simplified form. There are no more common factors between what remains in the numerator and denominator to cancel out.

What would happen if x = 1/2?

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When x = 1/2, the denominator 2x - 1 = 0, making the original expression undefined. Always remember that simplified expressions maintain the same domain restrictions as the original!

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