Mathematics
Roots of a Quadratic Equation
द्विघात समीकरण के मूल
Roots of a Quadratic Equation, taught in Class 10 Mathematics under the chapter Quadratic Equations, introduces the values of the variable that make a quadratic expression equal to zero. Students learn to identify roots, verify them by substitution, and connect the sum and product of the roots with the coefficients. The topic also helps them form a quadratic equation when its roots are known and use these relationships to solve and check mathematical problems.
Practice questions
If the equation \(x^2-2mx+(m^2-m)=0\) has real roots, what is the correct condition on \(m\)?ExpertLevel 32If the sum of the roots of a quadratic equation is 7 and their product is 10, which is its monic quadratic equation?ExpertLevel 32If the roots of a quadratic equation are reciprocals of each other and their sum is (\frac{5}{2}), which equation is possible?ExpertLevel 32If ((m-1)x^2+2(m+1)x+(m-1)=0) has real reciprocal roots, what is the correct condition on (m)?ExpertLevel 32If (\alpha,\beta) are the roots of (x^2-2(a+3)x+a^2+6a+5=0), what is the positive value of (\alpha-\beta)?ExpertLevel 32If one root of 2x² − (3p + 2)x + p(p + 2) = 0 is p, what will be the other root?MediumLevel 32If the positive difference between the roots of (5x^2-16x+p=0) is (\frac{6}{5}), what is the value of (p)?ExpertLevel 33If (x^2-2(a+5)x+a^2+18=0) has equal roots, what is the value of (a)?ExpertLevel 33If (\alpha,\beta) are the roots of (x^2-7x+12=0), which equation has roots (2\alpha+3) and (2\beta+3)?ExpertLevel 33If the roots of (3x^2+mx+10=0) are in the ratio (2:5), what are the possible values of (m)?ExpertLevel 33Which statement is always correct about x² − (a + 8)x + 8a = 0?MediumLevel 33If (\alpha,\beta) are the roots of (x^2-8x+2=0), what is (\frac{1}{\alpha^2}+\frac{1}{\beta^2})?ExpertLevel 33If (kx^2-12x+k=0) has real reciprocal roots, what is the correct condition on (k)?ExpertLevel 33If the roots of (x^2+px+24=0) are (r) and (r+2), what are the possible values of (p)?ExpertLevel 33For (16x^2-8(a-2)x+a^2-6a=0) to have real roots, what is the correct condition on (a)?ExpertLevel 33For (9x^2-6(a-1)x+a^2-4a-5=0) to have real roots, what is the correct condition on (a)?ExpertLevel 33What is the correct condition on \(a\) for the quadratic equation \(9x^2-6(a-1)x+a^2-4a-5=0\) to have real roots?ExpertLevel 33If (\alpha,\beta) are the roots of (x^2-9x+20=0), what is (\frac{\alpha+2}{\alpha-2}+\frac{\beta+2}{\beta-2})?ExpertLevel 33If (\alpha,\beta) are roots of (x^2-11x+30=0), what is the correct value of (\frac{\alpha+1}{\alpha-1}+\frac{\beta+1}{\beta-1})?ExpertLevel 33If (\alpha,\beta) are roots of (x^2-11x+30=0), what is (\frac{\alpha+1}{\alpha-1}+\frac{\beta+1}{\beta-1})?ExpertLevel 33