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 the equation \(x^2-2(a-1)x+(a^2+1)=0\) has no real roots, which condition on \(a\) is correct?If (a\neq0) and (D=b^2-4ac) is negative in (ax^2+bx+c=0), what is the correct statement about the graph and roots?What is the nature of the roots of the quadratic equation \(2x^2-6\sqrt{2}x+9=0\)?Choose the correct conclusion about the nature of the roots of the equation \(3x^2-4\sqrt{3}x+5=0\).What is the nature of the roots of the equation \\(5x^2-2\\sqrt{30}x+6=0\\)?If the roots of the equation \(x^2-2(k+2)x+(k^2+3k+7)=0\) are real, which condition on \(k\) is necessary?If x² − 2(k − 3)x + (k² − 8k + 20) = 0 has no real roots, what is the correct condition on k?For the equation \(x^2-(r+5)x+5r=0\) to have two real and distinct roots, which of the following conditions is correct?For the quadratic equation \(x^2-(s+4)x+4s=0\), what must be the value of \(s\) for its roots to be equal?If p ≠ 1 in (p − 1)x² − 2(p + 1)x + (p + 3) = 0, what is the condition on p for real roots?In (q + 2)x² − 2(q − 1)x + q = 0, with q ≠ −2, what is the value of q for equal roots?For the equation \\((q+2)x^2-2(q-1)x+q=0\\), which value of \\(q\\) gives equal roots, given that \\(q\\ne-2\\)?If x² + 2(t + 1)x + (3t + 7) = 0 has no real roots, in which interval will t lie?What is the correct condition on \(t\) for the equation \(x^2+2(t+1)x+(3t+7)=0\) to have no real roots?If (x^2-2(a+b)x+2ab=0) has real roots, which statement is always true for (a) and (b)?If a and b are real numbers, which is the correct condition for the equation \(x^2-2(a-b)x+(a+b)^2=0\) 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 \(ax^2+bx+c=0\)?What is the nature of the roots of the equation \\(7x^2-2\\sqrt{21}x+3=0\\)?Which option is correct about the roots of 2x² − 3√2 x + 7 = 0?If the equation \(x^2-2\mu x+2\mu=0\) has two real and distinct roots, which condition on \(\mu\) is correct?