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15 (a) Factorise $a^2 - b^2$ - Edexcel - GCSE Maths - Question 15 - 2018 - Paper 1

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15 (a) Factorise $a^2 - b^2$. (b) Hence, or otherwise, simplify fully $(x^2 + 4)^2 - (x^2 - 2)^2$.

Worked Solution & Example Answer:15 (a) Factorise $a^2 - b^2$ - Edexcel - GCSE Maths - Question 15 - 2018 - Paper 1

Step 1

Factorise $a^2 - b^2$

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Answer

The expression a2−b2a^2 - b^2 is a difference of squares, which can be factored using the formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).

Thus, the factorised form of a2−b2a^2 - b^2 is: (a−b)(a+b)(a - b)(a + b).

Step 2

Hence, or otherwise, simplify fully $(x^2 + 4)^2 - (x^2 - 2)^2$

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Answer

To simplify the expression (x2+4)2−(x2−2)2(x^2 + 4)^2 - (x^2 - 2)^2, we recognize that this is also a difference of squares:

Let:

  • A=x2+4A = x^2 + 4
  • B=x2−2B = x^2 - 2

The expression becomes: (A2−B2)=(A−B)(A+B)(A^2 - B^2) = (A - B)(A + B).

Calculating A−BA - B and A+BA + B:

  • A−B=(x2+4)−(x2−2)=4+2=6A - B = (x^2 + 4) - (x^2 - 2) = 4 + 2 = 6
  • A+B=(x2+4)+(x2−2)=2x2+2A + B = (x^2 + 4) + (x^2 - 2) = 2x^2 + 2

Thus, substituting back, we have: extSimplifiedresult=(6)(2x2+2)=12x2+12. ext{Simplified result} = (6)(2x^2 + 2) = 12x^2 + 12.

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