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 \(x^2-2rx+(r^2+25)=0\), where \(r\) is any real number, what is the correct conclusion about the nature of its roots?How many points of intersection are there between the parabola \(y=x^2-2kx+k^2+4\) and the x-axis?How will the parabola \(y=x^2-2(k-1)x+(k-1)^2\) meet the \(x\)-axis?For the parabola \(y=x^2-2kx+(k^2-9)\), where \(k\) is any real number, how will it intersect the \(x\)-axis?A geometric situation leads to the quadratic equation \(L^2-2(a+4)L+(a^2+6a+13)=0\). What condition on \(a\) is necessary for \(L\) to have real values?For the equation \(n^2-2pn+(p^2-7p)=0\), what condition on \(p\) is necessary for \(n\) to have two real and distinct roots?If \(a,b,c\) are integers and \(a\ne0\), which condition identifies that the quadratic equation \(ax^2+bx+c=0\) has two distinct irrational real roots?If the equation \(x^2-2(a+2)x+(a^2+6a+8)=0\) has no real roots, which condition on \(a\) is correct?For which values of \(k\) will the quadratic equation \(x^2-2(k+1)x+(k^2+1)=0\) have two distinct real roots?If the discriminant of a quadratic equation is ( D=-(p-1)^2 ) and ( p\ne 1 ), what is the nature of its roots?What is the nature of the roots of \(x^2-2(1+\sqrt{5})x+(6+2\sqrt{5})=0\)?If D = b² − 4ac is positive but not a perfect square, what is the most accurate nature of the roots of ax² + bx + c = 0?What is the nature of the roots of the equation \(6x^2-4\sqrt{6}x+4=0\)?Choose the correct conclusion about the nature of the roots of \(2x^2-5\sqrt{2}x+8=0\).What is the nature of the roots of the equation \(3x^2-2\sqrt{21}x+7=0\)?For a quadratic equation \(ax^2+bx+c=0\) with real coefficients, where \(a\ne0\), which condition identifies that its roots have opposite signs?If x² − 2(k − 4)x + (k² − 10k + 27) = 0 has no real roots, which condition on k is correct?Which condition is required for the equation \(x^2-(t+7)x+7t=0\) to have two real and distinct roots?For the quadratic equation \(x^2-(u+6)x+6u=0\) to have equal roots, what must be the value of \(u\)?If p ≠ 2 in (p − 2)x² − 2(p + 2)x + (p + 6) = 0, what is the condition on p for real roots?