Mathematics
Nature of Roots
मूलों की प्रकृति
In Class 10 Mathematics, Nature of Roots explains how to determine the type and number of solutions of a quadratic equation. Students use the discriminant, b² − 4ac, to identify whether an equation has two distinct real roots, two equal real roots, or no real roots. The topic connects algebraic calculations with the graph of a quadratic function and helps learners interpret equations, compare cases, and solve related problems from the chapter Quadratic Equations.
Practice questions
For which values of \(\mu\) will the roots of the quadratic equation \(x^2-2\mu x+2\mu=0\) be equal?If (\alpha\neq-2) in ((\alpha+2)x^2-2\alpha x+(\alpha-1)=0), what is the condition on (\alpha) for real roots?The equation \(x^2-2(3t+1)x+(5t^2+2t+4)=0\) has no real roots. What is the correct interval for \(t\)?If the roots of the equation \(x^2+2(m-4)x+(m^2-7m+14)=0\) are equal, what is the value of \(m\)?If the roots of the equation \(x^2+2(m-4)x+(m^2-7m+14)=0\) are real, which condition on \(m\) is correct?For equal roots in (2x^2-2(3k-1)x+(k+1)^2=0), what are the values of (k)?Which of the following quadratic equations has one positive root and one negative root?If x² − 2px + (p² − 25) = 0, what is the nature of the roots for any real value of p?If \(x^2-2px+(p^2+16)=0\), what will be the nature of its roots for any real value of \(p\)?Assertion: In the equation \(x^2-2(a-b)x+(a+b)^2=0\), if \(ab<0\), its roots are real and distinct. Reason: The discriminant of this equation is \(D=-16ab\). Choose the correct option.Assertion: The graph of the quadratic equation \(2x^2-4x+7=0\) does not intersect the \(x\)-axis. Reason: The discriminant of this equation is \(D=-40\). Choose the correct option.A student writes D=16 for x²−2(k−2)x+k²=0. What is the correct discriminant D?A student claims that the equation \(x^2+(m-4)x+m=0\) will have equal real roots only for \(m=4\), because the coefficient of \(x\) becomes zero. Which is the correct evaluation of this claim?If the discriminant of a quadratic equation is \(D=(w+2)^2\), what must be the value of \(w\) for the equation to have equal roots?If the discriminant of a quadratic equation is \(D=(z-4)(z+6)\), which interval of \(z\) results in no real roots of the equation?If the discriminant of a quadratic equation is \(D=12n-36\), what condition on \(n\) is necessary for the equation to have two real and distinct roots?If the discriminant of a quadratic equation is \(D=36-9h^2\), what values of \(h\) will give equal roots?What is the nature of the roots of the quadratic equation \(x^2+2(2-\sqrt{5})x+9=0\)?What is the nature of the roots of the equation \(x^2-2(2+\sqrt{3})x+(7+4\sqrt{3})=0\)?If x² − 2rx + (r² − 49) = 0, what is the nature of the roots for any real value of r?