Nature of Roots

First verify the reason by calculating the discriminant. In x²−6x+11=0, a=1, b=-6, and c=11. Therefore D=b²−4ac=(-6)²−4(1)(11)=36−44=-8. The reason is true. Since D<0, the quadratic equation has no real roots; its roots are a complex conjugate pair. Consequently, the assertion that the roots are real and irrational is false. The word irrational applies to non-real? No: irrational numbers are real numbers that cannot be expressed as a ratio of integers, so these complex roots are not called irrational roots. Thus option A correctly states that the assertion is false while the reason is true. The other choices either incorrectly accept the assertion or reject the correctly calculated discriminant.

Here, \\(a=8, b=2, c=-3\\). Therefore, the discriminant is \\(D=b^2-4ac=2^2-4(8)(-3)=100\\). Since \\(D>0\\), the roots are real and distinct; since \\(100\\) is a perfect square, they are also rational. In fact, the roots are \\(\frac{1}{2}\\) and \\(-\frac{3}{4}\\). Exam tip: If \\(D>0\\) and the discriminant is a perfect square, the roots are real, distinct, and rational.

Here, \(a=7\), \(b=3\), and \(c=2\). Thus, the discriminant is \(D=b^2-4ac=3^2-4(7)(2)=9-56=-47\). Since \(D<0\), the equation has no real roots. Option B would be correct only if \(D=0\), but the discriminant here is negative. Exam tip: To determine the nature of the roots, first calculate the sign of \(D=b^2-4ac\).

Here, \(a=12\), \(b=-12\), and \(c=3\). Therefore, the discriminant is \(D=b^2-4ac=(-12)^2-4(12)(3)=144-144=0\). When \(D=0\), the roots are real and equal. In fact, the equation can be written as \(3(2x-1)^2=0\), giving the repeated root \(x=\frac{1}{2}\). Option B is incorrect because \(D>0\) gives two distinct real roots. Exam tip: For a quadratic equation, \(D=0\) always indicates equal real roots.

The discriminant is \(D=b^2-4ac\). Equal real roots require \(D=0\), so \(b^2=4ac\). If \(D>0\), the roots are distinct real roots. Exam tip: determine the nature of roots from the sign of the discriminant first.

For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(D=b^2-4ac\). Here, \(a=3, b=1, c=4\), so \(D=1^2-4(3)(4)=1-48=-47\). Since \(D<0\), the equation has no real roots. Exam tip: \(D<0\) indicates no real roots, whereas \(D=0\) indicates two equal real roots.

The discriminant is \\(D=b^2-4ac=(-19)^2-4(10)(6)=121\\). Since \\(D>0\\), the roots are real and distinct. Also, \\(121\\) is a perfect square, so both roots are rational. In fact, the roots are \\(x=\frac{2}{5}\\) and \\(x=\frac{3}{2}\\). Therefore, option A is correct; option D is incorrect because the discriminant is a perfect square. Exam tip: Use \\(D>0\\) for distinct real roots, and check whether \\(D\\) is a perfect square to decide if they are rational.

Here, \(a=2\), \(b=-7\), and \(c=1\). Therefore, the discriminant is \(D=b^2-4ac=(-7)^2-4(2)(1)=41\). Since \(D>0\), the roots are real and distinct; because 41 is not a perfect square, they are also irrational. Option B is incorrect because rational roots in this test require a positive perfect-square discriminant. Exam tip: when \(D>0\) and is not a perfect square, the roots are real, distinct, and irrational.

For irrational distinct roots, (D>0) and (D) must not be a perfect square. (15) satisfies this condition.

The discriminant \(\Delta=144\) is a positive perfect square. Since \(\Delta>0\), the two roots are real and distinct; because the coefficients are rational and the discriminant is a perfect square, the roots are rational. Hence, option A is correct. Option B is true only when \(\Delta=0\). Exam tip: first use the sign of \(\Delta\) to determine whether the roots are real and distinct, and then check whether it is a perfect square to determine rationality.

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=1\), \(b=-2(k-5)\), and \(c=36\). Thus, \(D=4(k-5)^2-144=0\), which gives \((k-5)^2=36\). Therefore, \(k-5=\pm6\), so \(k=11\) or \(k=-1\). Exam tip: For equal roots of a quadratic equation, set the discriminant directly equal to zero.

For real roots, ((2k+1)^2-40\geq0) is needed. Hence (2k+1\leq-2\sqrt{10}) or (2k+1\geq2\sqrt{10}).

A quadratic equation has two real and equal roots when its discriminant satisfies \\(D=b^2-4ac=0\\). Here, \\(a=4, b=-9, c=q\\), so \\(D=(-9)^2-4(4)(q)=81-16q\\). Setting this equal to zero gives \\(q=\\frac{81}{16}\\). Exam tip: for equal roots, immediately use the condition \\(D=0\\).

For equal roots, the discriminant must be zero: \(D=b^2-4ac=0\). Here, \(a=p\), \(b=-18\), and \(c=9\), so \((-18)^2-4(p)(9)=0\), giving \(324-36p=0\) and hence \(p=9\). Exam tip: For equal roots, immediately apply the condition \(D=0\).

Here (D=4(k+1)^2-4(k+5)). (D\geq0) gives (k^2+k-4\geq0), so solve the resulting inequality carefully.

For the given quadratic equation, \(a=1\), \(b=-10\), and \(c=30\). Thus, the discriminant is \(D=b^2-4ac=(-10)^2-4(1)(30)=100-120=-20\). Since \(D<0\), the equation has no real roots. Option B would be correct only if \(D=0\). Exam tip: the sign of the discriminant directly determines the nature of the roots.

Here, \(a=1\), \(b=-21\), and \(c=110\). Thus, the discriminant is \(D=b^2-4ac=(-21)^2-4(1)(110)=441-440=1\). Since \(D>0\) and 1 is a perfect square, the equation has two distinct real rational roots. Therefore, option A is correct; equal roots occur only when \(D=0\). Exam tip: \(D>0\) indicates distinct real roots, and a perfect-square discriminant indicates that the roots are rational.

For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(D=b^2-4ac\). Here, \(a=1, b=12, c=37\), so \(D=12^2-4(1)(37)=144-148=-4\). Since \(D<0\), the equation has no real roots. The condition \(D=0\), given in option B, would instead indicate two equal real roots. In an exam, first check the sign of the discriminant to determine the nature of the roots.

Here, \(a=5, b=-14, c=11\). Therefore, the discriminant is \(D=b^2-4ac=(-14)^2-4(5)(11)=196-220=-24\). Since \(D<0\), the equation has no real roots. Option D results from a sign error that gives \(D=24\). Exam tip: \(D<0\) means no real roots, \(D=0\) means equal real roots, and \(D>0\) means distinct real roots.

Here, \(a=1\), \(b=-19\), and \(c=90\). Therefore, the discriminant is \(\Delta=b^2-4ac=(-19)^2-4(1)(90)=361-360=1\). Since \(\Delta\) is positive and a perfect square, the roots are real, rational, and distinct; in fact, they are 9 and 10. Option B would require \(\Delta=0\). Exam tip: \(\Delta>0\) indicates distinct real roots, and a perfect-square discriminant indicates that they are rational.