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 \(v\) is any real number, which statement correctly describes the nature of the roots of \(2x^2-4vx+2v^2+5=0\)?If the equation \(x^2+2gx+g^2-6g+11=0\) has real roots, what condition must \(g\) satisfy?For the quadratic equation \(x^2+2gx+g^2-6g+11=0\) to have equal roots, what should be the value of \(g\)?If the equation \(x^2-2hx+h^2+8h=0\) has real and distinct roots, which condition on \(h\) is necessary?If the quadratic equation \(x^2-2hx+h^2+8h=0\) has equal roots, what is the value of \(h\)?What condition on \(a\) is necessary for the equation \(3x^2-2(2a+1)x+(a^2+a+1)=0\) to have real roots?For which values of \(a\) will the equation \(3x^2-2(2a+1)x+(a^2+a+1)=0\) have real and distinct roots?For which interval of the parameter \(a\) does the quadratic equation \(3x^2-2(2a+1)x+(a^2+a+1)=0\) have no real roots?A quadratic equation with rational coefficients has discriminant D = 18. A student writes that its roots are rational. What is the correct correction?A student assumes that a quadratic equation has two real roots when its discriminant is D = -4. What is the correct conclusion?If the discriminant of a quadratic equation is (D=(a-1)^2+5), where (a) is a real number, which statement correctly describes the nature of its roots?If the discriminant of a quadratic equation is \(D=-(b+3)^2\), when will its roots be real and equal?For a quadratic equation, if the discriminant is \(D=-(b+3)^2\) and \(b\neq -3\), what will be the nature of its roots?What is the nature of the roots of x² − (2r+5)x + (r²+5r+4) = 0?Let \(r\) be a real number. Which statement correctly describes the nature of the roots of the equation \(x^2-(2r+5)x+(r^2+5r+7)=0\)?Which of the following quadratic equations has non-real roots for every real value of \(m\)?What is the nature of the roots of the quadratic equation \(x^2-2(k-3)x+k^2-6k+8=0\)?If x^2-2(k-3)x+k^2-6k+9=0 is a quadratic equation, what will be the nature of its roots for all real values of k?If \(n\) is any real number, what is the nature of the roots of the equation \(x^2+2(n+2)x+n^2+4n+1=0\)?For a real parameter \(n\), which statement correctly describes the nature of the roots of the equation \(x^2+2(n+2)x+n^2+4n+8=0\)?