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
If \(a\) is a real number, which statement correctly describes the nature of the roots of the equation \(3x^2-6ax+(3a^2+2)=0\)?HardLevel 37If 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?HardLevel 37If the two roots of (x^2-2(a+b)x+4ab=0) are equal, which of the following relations must be true?HardLevel 37If a and b are real constants, when will the two roots of \(x^2-(a+b)x+ab=0\) be equal?HardLevel 37If \(a\ne b\) and \(a,b\) are real numbers, what is the nature of the roots of the equation \(x^2-(a+b)x+ab=0\)?HardLevel 37What is the nature of the roots of the equation \\(x^2+2(a+1)x+a^2+2a+5=0\\)?HardLevel 37If the quadratic equation \(x^2-2mx+3m=0\) has real and equal roots, what are the possible values of \(m\)?HardLevel 37What condition on \(m\) is necessary for the equation \(x^2-2mx+3m=0\) to have two real and distinct roots?HardLevel 37For the equation \(x^2-2mx+3m=0\) to have no real roots, which is the correct interval for \(m\)?HardLevel 37If the roots of the equation \(x^2+2kx+k^2-4k+8=0\) are real, what is the correct condition on \(k\)?HardLevel 37If the equation \(x^2+2kx+k^2-4k+8=0\) has real and equal roots, what is the value of \(k\)?HardLevel 37For the equation \((k-2)x^2-2(k+1)x+k=0\) to have real and equal roots, what should be the value of \(k\)?HardLevel 37For which interval of a will x² + 2(a − 2)x + 2a + 5 = 0 have no real roots?HardLevel 37If the discriminant of a quadratic equation \(ax^2+bx+c=0\) with real coefficients is \(D<0\), what is the nature of its roots?MediumLevel 38What is the nature of the roots of the equation \(3x^2-10x+3=0\)?MediumLevel 38Choose the correct option regarding the nature of the roots of the quadratic equation \(5x^2-6x+2=0\).MediumLevel 38What is the nature of the roots of x² − 9x + 20 = 0?EasyLevel 38What is the nature of the roots of the quadratic equation \(x^2+6x+13=0\)?MediumLevel 38What is the repeated root of the equation \(4x^2-20x+25=0\)?MediumLevel 38Identify the nature of the roots of x² − 5x + 5 = 0.MediumLevel 38