2. (a) Expand and simplify $(7 + \sqrt{5})(3 - \sqrt{5})$ - Edexcel - A-Level Maths Pure - Question 4 - 2010 - Paper 2
Question 4
2. (a) Expand and simplify $(7 + \sqrt{5})(3 - \sqrt{5})$.
(b) Express $\frac{7 + \sqrt{5}}{3 + \sqrt{5}}$ in the form $a + b\sqrt{5}$, where $a$ and $b$ are intege... show full transcript
Worked Solution & Example Answer:2. (a) Expand and simplify $(7 + \sqrt{5})(3 - \sqrt{5})$ - Edexcel - A-Level Maths Pure - Question 4 - 2010 - Paper 2
Step 1
Expand and simplify $(7 + \sqrt{5})(3 - \sqrt{5})$
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Answer
To expand the expression, we can use the distributive property:
(7+5)(3−5)=7⋅3+7⋅(−5)+5⋅3+5⋅(−5)
Calculating each term gives:
7⋅3=21
7⋅(−5)=−75
5⋅3=35
5⋅(−5)=−5
Combining these results:
21−75+35−5
Next, we group the like terms:
21−5+(−75+35)=16−45
Thus, the final answer is:
16−45
Step 2
Express $\frac{7 + \sqrt{5}}{3 + \sqrt{5}}$ in the form $a + b\sqrt{5}$
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Answer
To express this fraction in the desired form, we can multiply the numerator and the denominator by the conjugate of the denominator:
3+57+5⋅3−53−5
This gives:
=(3+5)(3−5)(7+5)(3−5)
Calculating the denominator first:
(3+5)(3−5)=32−(5)2=9−5=4
Next, we can use the result from part (a) for the numerator:
