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 the equation \\((r+2)x^2-2(r+5)x+(r+2)=0\\), what is the value of \(r\) for which the roots are real and equal?HardLevel 38If m is a real number, when will the roots of \(x^2-2(m-4)x+m^2-16=0\) be real and distinct?HardLevel 38What condition on \(a\) is required for the equation \(x^2+2(a+3)x+a^2+10a+17=0\) to have no real roots?HardLevel 38If (x^2+2px+2p+9=0) has real roots, which condition on (p) is correct?HardLevel 38Which value of \(q\) is required for the two roots of the quadratic equation \(2x^2-(q+4)x+2q=0\) to be equal?HardLevel 38For which value of k will (k+3)x² + 2kx + (k−1) = 0 have equal roots?HardLevel 38For a real parameter \(t\), which value of \(t\) will make the equation \((t-1)x^2-2(t+2)x+(t+5)=0\) have equal roots?HardLevel 38What condition on \(c\) is necessary for the quadratic equation \(x^2-2(c+1)x+c^2-2c+5=0\) to have real roots?HardLevel 38What is the nature of the roots of the quadratic equation \(3x^2-2\sqrt{15}x+5=0\)?HardLevel 38How will the roots of (x^2-2\sqrt{7}x+3=0) be?HardLevel 38If the equation \(x^2-2\sqrt{n}x+4=0\) has real and equal roots, what is the value of \(n\)?HardLevel 38What is the correct condition for real roots of (5x^2-2(k+1)x+(k-3)=0)?HardLevel 38What is the correct statement about the real roots of x² + 2(2y−1)x + 4y² + 3 = 0?HardLevel 38What is the nature of roots of (x^2+2(3u+1)x+9u^2-6u+5=0)?HardLevel 38If the discriminant of a quadratic equation is \(D=(s-2)(s+5)\), where \(s\) is real, when will its roots be real and distinct?HardLevel 38If the discriminant of a quadratic equation is \(D=(z+1)^2-9\), in which interval must \(z\) lie for the equation to have no real roots?HardLevel 38For the quadratic equation \(x^2-(a+6)x+6a=0\) to have equal roots, what is the value of \(a\)?HardLevel 38If \(a\ne b\), what is the nature of the roots of the equation \(x^2-2(a+b)x+4ab=0\)?HardLevel 38When will the two roots of the equation \(x^2-(m+n)x+mn=0\) be equal?HardLevel 38Assume that \(a\) and \(b\) are real numbers. Which statement about the roots of the equation \(x^2+2(a-b)x+(a+b)^2=0\) is correct?HardLevel 38