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Verify quadratic fraction cancellations and exact values

Start a set of 8
1 OF 14
\(\displaystyle \frac{x^2-4}{x^2-x-2}\)

Which step preserves this fraction?

Year 10, Chapter 10, 4 difficulty levels, No calculatorBuild a worksheet

Factorisation, cancellation and exact checks

This placement practises simplifying single fractions. Numerical and simple-variable four-operation work remains in the earlier algebra chapter

  1. Record the original domain Every zero of the original denominator stays excluded, whether its factor cancels or remains
  2. Factor whole expressions Write sums as products before cancelling. A common factor must multiply the entire numerator and denominator \(\displaystyle x^2+2x-8=(x-2)(x+4)\)
  3. Cancel matching copies Cancel equal factors only when they are nonzero. Repeated factors cancel one matching copy at a time \(\displaystyle \begin{aligned}&\frac{(x-2)^2}{x-2}\\&=x-2,\quad x\ne2\end{aligned}\)
  4. Use the displayed simplified form For N(a) and D(a), retain the displayed coefficient and factor arrangement before evaluating. Their quotient gives the exact value

Worked example

Simplify, retain restrictions and evaluate at x = 1

\[\frac{6(x-2)(x+3)}{9(x-2)(x+1)}\]
  1. \(\displaystyle x\ne2,\quad x\ne-1\)Restrictions come from the original denominator
  2. \(\displaystyle \frac{\frac23(x+3)}{x+1}\)Reduce six ninths and cancel one factor of x minus two. Put the numerical coefficient in the numerator, as the practice display does
  3. \(\displaystyle N(1)=\frac23(1+3)=\frac83\)Use the displayed numerator
  4. \(\displaystyle D(1)=1+1=2\)Use the displayed denominator
  5. \(\displaystyle \frac83\div2=\frac43\)Divide and reduce exactly

Common mistakes

Dropping a restriction
✕\(\displaystyle x\ne-1\quad\text{only}\)✓\(\displaystyle x\ne-1,\quad x\ne2\)
Both original denominator zeros remain excluded in the worked example
Ignoring the scalar coefficient
✕\(\displaystyle \frac{6A}{9B}=\frac AB\)✓\(\displaystyle \frac{6A}{9B}=\frac{2A}{3B}\)
Cancel the numerical HCF 3, not unequal factors 6 and 9
Comparing different normalisations
✕\(\displaystyle N(1)=8,\quad D(1)=6\)✓\(\displaystyle N(1)=\frac83,\quad D(1)=2\)
Both quotients are four thirds, but the worked example asks for the numerator and denominator of the displayed simplified expression

Difficulty levels

introSimplify a single fraction with linear, quadratic or square-difference factor data; verify setup, result, original restrictions and an exact substitution
standardSimplify a single fraction with linear, quadratic or square-difference factor data; verify setup, result, original restrictions and an exact substitution
challengeSimplify a single fraction with linear, quadratic or square-difference factor data; verify setup, result, original restrictions and an exact substitution
examSimplify a single fraction with linear, quadratic or square-difference factor data; verify setup, result, original restrictions and an exact substitution

Questions about this topic

Are multiplication and addition missing?+

This topic is a factorisation-and-cancellation extension. The earlier algebra chapter covers the four operations on simple algebraic fractions

Why can a simplified numerator or denominator differ from mine?+

Equivalent fractions can be rescaled. The practice table asks for the particular numerator and denominator displayed in the correct result option, so use that normalisation

Can one substituted value prove the identity?+

No. Factorisation and valid cancellation establish it on the original domain. Substitution is a useful check, and a disagreement disproves a proposed identity

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs

In this chapter

Expand binomial products and special products10A Expanding and special products: Verify binomial and special-product expansions10B Common factors and difference of squares: Factorise common factors and differences of squares10B Common factors and difference of squares: Verify common-factor and square-difference products10C Quadratic factorisation and fractions: Factorise a quadratic expression completelySimplify an algebraic fraction by factorising and cancelling10C Quadratic factorisation and fractions: Verify monic and simple non-monic factorisationsVerify quadratic fraction cancellations and exact valuesSolve a quadratic equation by taking square rootsSolve a quadratic equation using the null factor lawSet up and solve a quadratic application problemIdentify and interpret parabola features10G Parabola features and transformations: Distinguish linear and quadratic relationsRead and use circle equations and graphsRead and use exponential growth and decay graphs10H Circles and exponential graphs: Classify a relation as linear, quadratic, circle or exponentialConnect reciprocal rules, tables and graphs