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
Which statement is correct for 4x² − 4x + 1 = 0?EasyLevel 37What is the nature of the roots of the equation \(x^2+5x+7=0\)?MediumLevel 37What is the nature of the roots of 2x² + 3x − 5 = 0?EasyLevel 37If \(a\ne0\) in the quadratic equation \(ax^2+bx+c=0\), and its roots are real and equal, which of the following conditions is correct?HardLevel 37For a quadratic equation \(ax^2+bx+c=0\) with integer coefficients, where \(a\ne0\), which condition on the discriminant \(D=b^2-4ac\) identifies two distinct irrational roots?HardLevel 37What is the correct condition on \\(k\\) for the roots of the equation \\(x^2-2(k+1)x+k^2=0\\) to be real and distinct?HardLevel 37What condition on \(m\) is required for the equation \(x^2+2(m-1)x+(m+5)=0\) to have no real roots?HardLevel 37For the equation \\(x^2-2(k-2)x+(k^2-4)=0\\) to have equal roots, what is the value of \\(k\\)?HardLevel 37What is the nature of the roots of the equation \(3x^2+2\sqrt{3}x+1=0\)?HardLevel 37Choose the nature of roots of (x^2-2\sqrt{5}x+1=0).HardLevel 37If the quadratic equation \(2x^2+px+8=0\) has real and equal roots, what are the possible values of \(p\)?HardLevel 37If \\(p\ne2\\), what must be the value of \\(p\\) for the equation \\((p-2)x^2+4x+1=0\\) to have equal roots?HardLevel 37What is the correct condition on \(k\) for the equation \(kx^2-6x+k=0\) to have real and distinct roots?HardLevel 37For which value of \(a\) will the equation \(x^2-(a+3)x+3a=0\) have equal roots?HardLevel 37For the two roots of the equation \(x^2-(r+4)x+4r=0\) to be equal, what must be the value of \(r\)?HardLevel 37What condition on the parameter \(a\) is necessary for the equation \(x^2+2(a+1)x+a^2+3=0\) to have real roots?HardLevel 37For the equation \(x^2-2(a-3)x+a^2-9=0\) to have real and equal roots, what should be the value of \(a\)?HardLevel 37For which condition on \(s\) will the equation \(x^2-2sx+s+2=0\) have real and distinct roots?HardLevel 37For which interval of (t) will the equation (x^2+2tx+t+4=0) have no real roots?HardLevel 37Why are the roots of (2x^2-(3q+1)x+q=0) always real and distinct?HardLevel 37