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15 (a) Factorise $a^2 - b^2$ (b) Hence, or otherwise, simplify fully $(x^2 + 4)^2 - (x^2 - 2)^2$ - 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-$(x^2-+-4)^2---(x^2---2)^2$-Edexcel-GCSE Maths-Question 15-2018-Paper 1.png

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 $a^2 - b^2$ (b) Hence, or otherwise, simplify fully $(x^2 + 4)^2 - (x^2 - 2)^2$ - Edexcel - GCSE Maths - Question 15 - 2018 - Paper 1

Step 1

Factorise $a^2 - b^2$

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Answer

To factorise the expression a2−b2a^2 - b^2, we can use the difference of squares formula, which states that:

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
Thus, the factorised form 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

We can apply the difference of squares formula again. Let:
A=(x2+4)A = (x^2 + 4)
B=(x2−2)B = (x^2 - 2)
Then we have:
(x2+4)2−(x2−2)2=(A−B)(A+B)(x^2 + 4)^2 - (x^2 - 2)^2 = (A - B)(A + B)
Now, calculate A−BA - B and A+BA + B:

  1. Calculate A−BA - B:
    A−B=(x2+4)−(x2−2)=4+2=6A - B = (x^2 + 4) - (x^2 - 2) = 4 + 2 = 6

  2. Calculate A+BA + B:
    A+B=(x2+4)+(x2−2)=2x2+2A + B = (x^2 + 4) + (x^2 - 2) = 2x^2 + 2

Now substituting back into the factorised form:
=(6)(2x2+2)=(6)(2x^2 + 2)
Factoring 22 out of the second term gives:
=12(x2+1)= 12(x^2 + 1)
Hence, the final simplified form is:

12(x2+1)12(x^2 + 1)

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