prove root 2 minus root 5 as irrerational
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let √2-√5 be a rational number x if possible
x=√2-√5
x²=(√2-√5)². (squaring both sides)
x²=(2+5-2√10)
x²=7-2√10
2√10=7-x²
√10=(7-x²)/2
we know that when we divide 2 rational number the quotient is rational
so , 7 is rational , x is rational so x² will also be rational , 2 is rational then , √10 is rational
but √10 is an irrational number so we arrive at a contradiction and this tells us that our supposition was wrong.
so √2-√5 is irrational
hope this helps
x=√2-√5
x²=(√2-√5)². (squaring both sides)
x²=(2+5-2√10)
x²=7-2√10
2√10=7-x²
√10=(7-x²)/2
we know that when we divide 2 rational number the quotient is rational
so , 7 is rational , x is rational so x² will also be rational , 2 is rational then , √10 is rational
but √10 is an irrational number so we arrive at a contradiction and this tells us that our supposition was wrong.
so √2-√5 is irrational
hope this helps
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