Nature of Roots

For a quadratic equation \(ax^2+bx+c=0\) to have two real and distinct roots, its discriminant must satisfy \(\Delta=b^2-4ac>0\). Here, \(a=1, b=-6, c=k\), so \(\Delta=(-6)^2-4(1)(k)=36-4k\). Thus, \(36-4k>0\), which gives \(k<9\). When \(k=9\), the roots are equal, and when \(k>9\), the roots are not real. Exam tip: for distinct real roots, always require \(\Delta>0\).

A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(D=b^2-4ac=2^2-4(1)(k)=4-4k\). Thus, \(4-4k<0\), which gives \(k>1\). If \(k=1\), then \(D=0\), so the equation has two equal real roots; hence option B is incorrect. Exam tip: Check the sign of the discriminant first to determine the nature of the roots.

The equation \(x^2-8x+16=0\) can be written as \((x-4)^2=0\). Thus, \(x-4=0\), so both equal roots are \(4\). The option \(-4\) is a common sign-error distractor. Exam tip: for equal roots, use \(x=-\frac{b}{2a}\) or check that the discriminant \(D=b^2-4ac=0\).

Here (D=9-4=5), which is positive but not a perfect square. So roots are real distinct and irrational.

For the given quadratic equation, \(a=2\), \(b=-5\), and \(c=2\). Therefore, the discriminant is \(D=b^2-4ac=(-5)^2-4(2)(2)=25-16=9\). Since \(D\) is positive and a perfect square, the roots are real, rational, and distinct. Option B is incorrect because a positive discriminant gives irrational roots only when it is not a perfect square. Exam tip: \(D>0\) indicates distinct real roots, while a perfect-square discriminant indicates that the roots are rational.

For the equation \(3x^2-12=0\), \(a=3\), \(b=0\), and \(c=-12\). Thus, the discriminant is \(D=b^2-4ac=0-4(3)(-12)=144\). Since \(D>0\), the roots are real and distinct; because \(\sqrt{D}=12\) is an integer, both roots are rational. In fact, the roots are \(x=2\) and \(x=-2\). Exam tip: If \(D>0\) and is a perfect square, the roots are real, rational, and distinct.

Here, \(a=5\), \(b=2\), and \(c=1\). Thus, the discriminant is \(D=b^2-4ac=2^2-4(5)(1)=4-20=-16\), so \(D<0\). A negative discriminant means that the quadratic equation has no real roots; its roots are complex conjugates. Option B would be correct only if \(D=0\). Exam tip: remember that \(D<0\), \(D=0\), and \(D>0\) correspond respectively to no real roots, equal real roots, and distinct real roots.

When a parabola touches the x-axis at exactly one point, the quadratic has one repeated real root. Hence, the discriminant (D=b^2-4ac=0), and the two roots are real and equal. If (D>0), the graph cuts the x-axis at two points, so option B is incorrect. Exam tip: touching means (D=0), cutting means (D>0), and staying away means (D<0).

Each intersection of the parabola with the x-axis represents one real root of the corresponding quadratic equation. Two distinct intersections therefore indicate two real and distinct roots, so the discriminant is \(D>0\). The condition \(D=0\) applies only when the graph touches the x-axis at one point. Exam tip: Use the number of x-intercepts to identify the nature of the roots quickly.

The points where the parabola intersects the x-axis represent the real roots of the quadratic equation. If it neither intersects nor touches the x-axis, there is no real intercept; therefore, the discriminant satisfies \(D<0\), and the equation has no real roots. In contrast, touching the x-axis would give \(D=0\).

For equal roots (D=0), so ((-8)^2-4(2)c=0) gives (c=8). Be careful with signs while substituting coefficients.

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=4\), \(b=m\), and \(c=1\), so \(D=m^2-16=0\). Therefore, \(m^2=16\), giving \(m=4\) or \(m=-4\). Choosing only \(m=4\) is incomplete because the negative value also produces equal roots. Exam tip: For equal-root questions, set \(b^2-4ac=0\) directly.

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=p\), \(b=4\), and \(c=1\), so \(D=4^2-4(p)(1)=0\), giving \(16-4p=0\) and hence \(p=4\). Option B has the wrong sign, while \(p=4\) also satisfies \(p\neq 0\), keeping the equation quadratic. Exam tip: For equal roots, immediately use \(D=0\).

Here \(a=3\), \(b=2k\), and \(c=k\). Therefore, the discriminant is \(D=b^2-4ac=(2k)^2-4(3)(k)=4k(k-3)\). For real roots, \(D\geq 0\), so \(k(k-3)\geq 0\), which gives \(k\leq 0\) or \(k\geq 3\). In option B, the discriminant is negative, so the roots are not real. Exam tip: For questions about real roots, begin by applying \(D\geq 0\).

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here, \(a=k\), \(b=-6\), and \(c=1\), so \(D=(-6)^2-4(k)(1)=0\). Thus, \(36-4k=0\), giving \(k=9\). The value 6 does not make the discriminant zero. Exam tip: For equal roots of a quadratic equation, set \(D=0\).

A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=-(k+1)\), and \(c=4\), so \(D=(k+1)^2-16\). Thus, \((k+1)^2<16\), which gives \(-4<k+1<4\), and hence \(-5<k<3\). Therefore, option A is correct. At the boundary values \(k=-5\) and \(k=3\), \(D=0\), so the equation has two equal real roots rather than no real roots. Exam tip: for ‘no real roots’, always use the condition \(D<0\).

Here, \(a=1\), \(b=2(k+1)\), and \(c=k^2\). Therefore, the discriminant is \(D=b^2-4ac=4(k+1)^2-4k^2=4(2k+1)\). Two distinct real roots require \(D>0\), so \(4(2k+1)>0\), which gives \(k>-rac{1}{2}\). At \(k=-\frac{1}{2}\), \(D=0\), so the roots are equal rather than distinct. Exam tip: use \(D>0\) specifically for two distinct real roots.

Here \(a=1, b=-2, c=5\), so \(D=b^2-4ac=4-20=-16\). A negative discriminant means there are no real roots. In exams, evaluate the complete \(b^2-4ac\), not \(b^2\) alone.

The discriminant D = 49 is positive and a perfect square. Since D > 0, the roots are real and distinct; because D is a perfect square, the roots are rational. Therefore, option A is correct. The roots would be equal only when D = 0, so option B is incorrect. Exam tip: If D > 0 and is a perfect square, the roots are real, rational, and distinct.

For a quadratic equation, D = b² − 4ac. Since D = 12 is positive, the roots are real and distinct. Also, 12 is not a perfect square and the coefficients are rational, so the roots are irrational. Hence, option A is correct. Exam tip: D > 0 gives distinct real roots, D = 0 gives equal roots, and D < 0 gives no real roots.