Is it true that there exists a quadratic equation whose coefficients are rational but both of its roots are irrational?

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True

False

answer is A.

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Detailed Solution

Given statement is that there exists a quadratic equation whose coefficients are rational but both of its roots are irrational.
Consider the example of the equation

x 2 − 4 x − 3 = 0

. It has rational coefficients

1, - 4

and

- 3

.
The quadratics formula to determine the roots of the equation is,

x = − b ± b 2 − 4 a c 2 a = − − 4 ± − 4 2 − 41 − 3 21 = 4 ± 16 + 12 2 = 2 ± 7

Both the roots

2 + 7

and

2 − 7

are irrational in nature because

7

is irrational.
Thus it is true that a quadratic equation having coefficients as the rational number can have irrational roots.
Hence, the given statement is true.

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