Irrational numbers

Substituting r = 2 gives the quadratic x² − 4x + 2. Its discriminant is D = (−4)² − 4(1)(2) = 16 − 8 = 8. A positive discriminant means that there are two distinct real zeroes. Since 8 is not a perfect square, √8 is irrational, so the zeroes are irrational. In fact, the quadratic formula gives x = [4 ± √8]/2 = 2 ± √2, which verifies both properties directly. Therefore the statement is true and option A is correct. Option B contradicts the nonsquare discriminant, option C contradicts D > 0, and option D is unnecessary because all required information is provided.

To determine the zeroes, set x² − k = 0. This gives x² = k and hence x = ±√k. For the zeroes to be real, k must be non-negative; for them to be irrational, k must be positive and not a perfect square. If k is a positive perfect square, √k is an integer and therefore rational. If k = 0, both zeroes are 0, which are rational and equal. If k is negative, the square root is not real. Consequently, the necessary and sufficient condition is that k be positive but not a perfect square. Thus option B is correct; the other options each violate either reality or irrationality.

For p(x) = x² + 6x + 6, identify a = 1, b = 6, and c = 6. The discriminant is D = b² − 4ac = 6² − 4(1)(6) = 36 − 24 = 12. Since D is positive, the polynomial has two distinct real zeroes. Since 12 is not a perfect square, √12 is irrational, so the zeroes are irrational. The quadratic formula gives x = (−6 ± √12)/2 = −3 ± √3, confirming both their reality and irrationality. Thus option A is correct. Option B would require a perfect-square discriminant, option C would require a negative discriminant, and option D would require D = 0. None of those conditions applies.

Rationalization removes a radical from a denominator by multiplying the numerator and denominator by the denominator’s conjugate. The conjugate of √5 − 2 is √5 + 2. Therefore, 1/(√5 − 2) = (√5 + 2)/[(√5 − 2)(√5 + 2)]. Applying the difference-of-squares identity gives the denominator 5 − 4 = 1, so the fraction simplifies to √5 + 2. Hence option A is correct. Option B is merely the original denominator expression and does not rationalize the fraction. Option C introduces an unsupported factor of 9, while option D is not equivalent to the original fraction. The essential step is choosing the conjugate, which converts the radical denominator into a rational number.