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Mathematics

One root of a quadratic equation is 3 + 2\sqrt{2}.

Statement (1): The other root of the given quadratic equation is 3 - 2\sqrt{2}.

Statement (2): If one root of the given quadratic equation is in the form of a surd, the other root is its conjugate.

  1. Both the statements are true.

  2. Both the statements are false.

  3. Statement 1 is true, and statement 2 is false.

  4. Statement 1 is false, and statement 2 is true.

Quadratic Equations

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Answer

Both the statements are true.

Reason

We are given that one root of the quadratic equation is 3 + 2\sqrt{2}. If the quadratic equation has real coefficients, the conjugate of a root that involves a surd (i.e., a square root or irrational number) must also be a root of the quadratic equation.

Therefore, 3 - 2\sqrt{2} is other root of the given quadratic equation.

So, statement (1) is true.

The property of conjugate roots holds for quadratic equations with real coefficients, meaning if one root involves a surd, the other root will be its conjugate. In this case, since 3 + 2\sqrt{2} is a surd, the other root must be 3 - 2\sqrt{2}.

So, statement (2) is true.

Hence, option 1 is correct.

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