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
What condition on \(u\) is necessary for the equation \(x^2-2ux+u^2+7u=0\) to have real roots?For which value of (m) will (m x^2+(m+3)x+3=0) have equal roots and remain a quadratic equation?For the quadratic equation \((a-1)x^2+2ax+(a+3)=0\) to have equal roots, what is the value of \(a\)?What is the correct condition on λ for real roots of x² + 2(λ−1)x + λ² + 1 = 0?If the roots of the equation \(x^2-2(\alpha+2)x+\alpha^2=0\) are real and distinct, what is the correct condition on \(\alpha\)?For the quadratic equation \(x^2+(2h-1)x+h^2=0\) to have equal roots, what is the value of \(h\)?For which value of \(n\) will the equation \(x^2+(n+2)x+2n=0\) have equal roots?What condition on k is required for the equation 4x² − 4(k + 1)x + (k² + 2k) = 0 to have real roots?For any real constant \(k\), what is the nature of the roots of the equation \(9x^2-6(k-1)x+(k-1)^2=0\)?Which equation gives real, irrational and distinct roots?If the discriminant of a quadratic equation is (D=(m-5)^2), which value of (m) is required for its roots to be equal?If the discriminant of a quadratic equation with real coefficients is \(D=2r^2+3\), where \(r\) is a real number, what is the correct conclusion about its roots?If ctc is a real number and the discriminant of a quadratic equation is c(D=-(t^2+4))c, how many real roots will the equation have?If \(a,b,c\) are real numbers and \(a\ne0\), which situation about the roots of the quadratic equation \(ax^2+bx+c=0\) is impossible?If \(p\) is any real number, what will be the nature of the roots of the equation \(x^2+2px+p^2-1=0\)?If \(p\) is a real constant, what is the nature of the roots of the equation \(x^2-2px+p^2=0\)?Which statement correctly describes the nature of the roots of \(x^2+2px+p^2+1=0\)?If \(a\) is a real number, what is the nature of the roots of the equation \(2x^2-4ax+(2a^2-3)=0\)?If \(a\) is a real number, which statement correctly describes the nature of the roots of the equation \(3x^2-6ax+(3a^2+2)=0\)?If the quadratic equation \(x^2-2(a+b)x+(a-b)^2=0\), where a and b are real numbers, has equal roots, which relation holds between a and b?