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Mathematics · 07 / 23

Expand, factor and recognise identities

Middle school · Worked example and 4 exercises with solutions, at your pace.

Need a foundation?Fractions: share the same wholePowers and roots: count the factors
UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

Expansion turns products into sums by distribution. Factoring reverses it using a common factor or identity. (a+b)² = a²+2ab+b² and (a−b)(a+b)=a²−b² hold for all real values. Choose the form that helps calculation, solving a zero product or sign analysis.

(x+3)*(x-3)=x^2-9
THE PICTURE TO REMEMBERExpand, factor and recognise identities
  1. 01Recognise form
  2. 02Factor or identity
  3. 03Transform
  4. 04Expand to check

WORKED EXAMPLE

From principle to reasoning

3x+6 = 3(x+2). x²−9 = (x−3)(x+3). Expanding (x+2)² gives x²+4x+4, including the cross terms.

Solve a problem in this subject

  1. Define the objects

    State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.

  2. Choose a transformation

    Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.

  3. Show the reasoning

    Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.

  4. Test and conclude

    Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.

YOUR TURN

01 · Try before checking

Factor 5x²+10x.

One hint

Both terms contain 5x.

Reveal and understand the solution

5x²+10x = 5x(x+2). Expanding again reproduces the original terms.

Compare method, conditions and answer; identify the exact part to revisit.

IMPROVE THROUGH PRACTICE

Three additional challenges with solutions

Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.

Reflect after practising
02 · Recognise and understandLink to this exercise

Expand (x + 4)².

A hint to get started

A squared sum includes a cross term.

Step-by-step solution
  1. (x + 4)(x + 4) gives x² + 4x + 4x + 16.
  2. Combine to obtain x² + 8x + 16.
  3. x² + 16 would omit 8x.

Which step did you miss? Retry without help, then explain why the method works.

03 · Apply the methodLink to this exercise

Factor 6x² − 9x.

A hint to get started

Identify a common factor in both terms.

Step-by-step solution
  1. The greatest common factor is 3x.
  2. 6x² − 9x = 3x(2x − 3).
  3. Expanding checks each original term.

Which step did you miss? Retry without help, then explain why the method works.

04 · Go furtherLink to this exercise

Solve x² − 16 = 0 without a discriminant.

A hint to get started

Factor a difference of squares.

Step-by-step solution
  1. x² − 16 = (x − 4)(x + 4).
  2. Either factor can be zero.
  3. The solutions are −4 and 4, not just 4.

Which step did you miss? Retry without help, then explain why the method works.

The trap to avoid

(a+b)² is not a²+b². Dividing by x can lose a zero solution.

Without notes, can you…

  • Explain the principle in your own words?
  • Rebuild the diagram and its connections?
  • Solve the exercise and justify the method?

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