Nature of Roots

For a quadratic equation 8ax^2+bx+c=0\u00029, the discriminant is 8D=b^2-4ac\u00029. Here, 8a=1, b=-2, c=-1\u00029, so 8D=(-2)^2-4(1)(-1)=4+4=8\u00029. Since 8c=-1\u00029, subtracting 8-4ac\u00029 results in adding 4; therefore, 4 is not the discriminant. Exam tip: identify the signs of 8a,b,c\u00029 before substitution.

Here, \(a=1, b=-2, c=-1\). Therefore, the discriminant is \(D=b^2-4ac=(-2)^2-4(1)(-1)=8\). Since \(D>0\), the roots are real and distinct. Also, 8 is not a perfect square, so the roots are irrational. Exam tip: If \(D>0\) and \(D\) is not a perfect square, the roots are real, irrational, and distinct.

For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(D=b^2-4ac\). Here, \(a=5\), \(b=-20\), and \(c=20\), so \(D=(-20)^2-4(5)(20)=400-400=0\). Therefore, the correct answer is 0. When \(D=0\), the two roots are equal. Exam tip: Always substitute the signed value of \(b\) before squaring it.

Here, \(a=5, b=-20, c=20\). The discriminant is \(D=b^2-4ac=(-20)^2-4(5)(20)=400-400=0\). Therefore, the roots are real and equal. In fact, the equation becomes \(5(x-2)^2=0\), so both roots are \(x=2\). Exam tip: \(D=0\) gives real and equal roots, whereas \(D>0\) gives distinct real roots.

Here, \(D=25>0\), so the two roots are real and distinct. Since 25 is a perfect square and the coefficients are rational, the roots are also rational. Option B would be correct only when \(D=0\), while irrational roots occur when the positive discriminant is not a perfect square. Exam tip: \(D>0\) gives distinct real roots, \(D=0\) gives equal real roots, and \(D<0\) gives non-real roots.

For a quadratic equation, D>0 means that the roots are real and distinct. Here, D=5>0, so the roots are real and distinct. Since 5 is not a perfect square and the coefficients are integers, the roots are also irrational. Therefore, option A is correct. Exam tip: D>0 gives real and distinct roots, D=0 gives equal roots, and D<0 gives non-real roots.

For a quadratic equation, the discriminant is \(D=b^2-4ac\). If \(D<0\), the equation has no real roots. Here, \(D=-9<0\), so option A is correct. Option B requires \(D>0\), while option C requires \(D=0\). Exam tip: determine the nature of the roots by checking the sign of the discriminant.

For equal roots, the discriminant must be zero. Here, \(a=1\), \(b=k\), and \(c=16\), so \(D=b^2-4ac=k^2-4(1)(16)=k^2-64\). Thus, \(k^2-64=0\), giving \(k^2=64\). Exam tip: For equal roots, immediately apply \(b^2-4ac=0\); do not confuse the constant term 16 with the required value of \(k^2\).

For \(ax^2+bx+c=0\), the product of the roots is \(c/a\). In option C, \(c/a=-6\), so one root is positive and the other is negative. Option A has a positive product, so its roots cannot have opposite signs. Exam tip: check whether \(c/a<0\).

For a quadratic equation \(ax^2+bx+c=0\), the roots are real and distinct when the discriminant \(D=b^2-4ac\) is positive. Here, \(a=1\), \(b=p\), and \(c=6\), so \(D=p^2-24\). Therefore, \(p^2-24>0\), which gives \(p^2>24\). If \(p^2=24\), the roots are real but equal. Exam tip: remember that real and distinct roots require \(D>0\).

A quadratic equation has no real roots when its discriminant \(D=b^2-4ac\) is negative. Here, \(a=1\), \(b=q\), and \(c=10\), so \(D=q^2-40\). Therefore, \(q^2-40<0\), which gives \(q^2<40\). Remember that \(q^2=40\) gives equal real roots, not no real roots.

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=m\), \(b=6\), and \(c=3\), so \(6^2-4(m)(3)=0\). Thus, \(36-12m=0\), giving \(m=3\). Therefore, option A is correct. Exam tip: For equal-root questions, immediately apply the condition \(D=0\).

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=2\), \(b=n\), and \(c=8\), so \(n^2-4(2)(8)=0\), giving \(n^2=64\). Remember that the question asks for \(n^2\), not \(n\).

For option A, the discriminant is \(D=b^2-4ac=(-9)^2-4(1)(20)=1\). Since \(D>0\) and 1 is a perfect square, the roots are real, rational, and distinct. In fact, the equation factors as \((x-4)(x-5)=0\), giving roots 4 and 5. The discriminants of options B, C, and D are \(-3,-39,-16\), respectively, so their roots are not real. Exam tip: For a quadratic equation, \(D>0\) and a perfect-square discriminant indicate distinct real rational roots.

For option B, \(a=1, b=-2, c=-2\). Therefore, the discriminant is \(\Delta=b^2-4ac=(-2)^2-4(1)(-2)=12\). Since \(\Delta>0\), the roots are real and distinct; since 12 is not a perfect square, the roots are irrational. In fact, the roots are \(1\pm\sqrt{3}\). Option A has discriminant 16, giving rational roots; option C has discriminant 0, giving equal roots; and option D has a negative discriminant, so it has no real roots. Exam tip: real, distinct, irrational roots require \(\Delta>0\) and \(\Delta\) not to be a perfect square.

In a quadratic equation, (D=0) means both roots are equal. This is a direct rule of nature of roots.

For a quadratic equation, the discriminant is \(D=b^2-4ac\). When \(D<0\), the equation has no real roots, so the statement claiming two real and distinct roots is false. Two real and distinct roots occur when \(D>0\), while \(D=0\) gives equal real roots. Exam tip: Always check the three cases \(D>0\), \(D=0\), and \(D<0\) to determine the nature of the roots.

The statement is incorrect. For a quadratic equation, \(D>0\) gives two real and distinct roots, not equal roots. Equal roots occur only when \(D=0\), while \(D<0\) gives no real roots. Exam tip: remember the sequence—\(D>0\): real and distinct roots, \(D=0\): equal roots, \(D<0\): non-real roots.

Here, \(a=4\), \(b=-4\), and \(c=3\). The discriminant is \(\Delta=b^2-4ac=(-4)^2-4(4)(3)=16-48=-32<0\). Therefore, the equation has no real roots. Option D is incorrect because irrational and distinct roots are real roots and require a positive discriminant. Exam tip: For a quadratic equation, \(\Delta<0\) means that it has no real roots.

In the standard form \(ax^2+bx+c=0\), \(a=6\), \(b=-5\), and \(c=1\). Thus, \(D=b^2-4ac=(-5)^2-4(6)(1)=25-24=1\), so option A is correct. Since \(D>0\), the roots are real and distinct; because \(D\) is also a perfect square, they are rational. Exam tip: always place the signed value of \(b\) in parentheses when calculating the discriminant.