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 \((q+3)x^2-2(q-2)x+q=0\), what is the value of \(q\) for which the roots are equal, given that \(q\ne -3\)?If (x^2+2(v+2)x+(4v+11)=0) has no real roots, in which interval will (v) lie?What is the correct condition on \(v\) for the equation \(x^2+2(v+2)x+(4v+11)=0\) to have no real roots?If \(a\) and \(b\) are real numbers and the roots of \(x^2-2(a+b)x+3ab=0\) are real, which statement about \(a\) and \(b\) is correct?If \(a\) and \(b\) are real numbers, which condition is necessary and sufficient for the equation \(x^2-2(a-2b)x+(a+2b)^2=0\) to have real roots?If \(a<0\), \(c>0\), and \(b\) is any real number, what will be the nature of the roots of the quadratic equation \(ax^2+bx+c=0\)?If \(a<0\), \(c<0\), and \(b=0\), what is the nature of the roots of the quadratic equation \(ax^2+c=0\)?What is the nature of the roots of \\(8x^2-4\sqrt{10}x+5=0\\)?Which option is correct about the roots of 5x² − 3√5x + 6 = 0?If the equation \(x^2-2\theta x+3\theta=0\) has two real and distinct roots, which condition on \(\theta\) is correct?For which values of \(\theta\) does the quadratic equation \(x^2-2\theta x+3\theta=0\) have equal roots?If (\alpha\neq-3) in ((\alpha+3)x^2-2\alpha x+(\alpha-2)=0), what is the condition on (\alpha) for real roots?The equation \(x^2-2(4t+1)x+(7t^2+2t+5)=0\) has no real roots. Which is the correct interval for \(t\)?If the quadratic equation \(x^2+2(m-5)x+(m^2-9m+24)=0\) has equal roots, what is the value of \(m\)?If m is a real number and the equation x² + 2(m − 5)x + (m² − 9m + 24) = 0 has no real roots, which condition on m is correct?If \(p\) is any real number, what is the nature of the roots of the equation \(x^2+2px+(p^2+1)=0\)?If x² − 2px + (p² − 64) = 0, what is the nature of the roots for any real value of p?For the quadratic equation \(x^2-2px+(p^2+36)=0\), what is the correct conclusion about the nature of its roots for any real value of \(p\)?Assertion: In the equation \(x^2-2(a-3b)x+(a+3b)^2=0\), if \(ab<0\), then its roots are real and distinct. Reason: The discriminant of this equation is \(D=-48ab\). Choose the correct option.Assertion: The graph of the parabola \(y=3x^2-6x+11\) does not intersect the \(x\)-axis. Reason: The discriminant of the corresponding quadratic equation is \(D=-96\). Choose the correct option.