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 \(kx^2-2(k+1)x+(k+4)=0\) has equal roots, what is the value of \(k\)?HardLevel 33What is the correct condition on \(a\) for the equation \(x^2-2(a+1)x+a^2+3=0\) to have real roots?HardLevel 33The roots of x^2-px+36=0 are positive integers and their difference is 5. What is p?MediumLevel 33If (\alpha,\beta) are the roots of (3x^2-10x+3=0), which equation has roots (\frac{1}{\alpha},\frac{1}{\beta})?HardLevel 33If one root of (x^2+(k-3)x+k=0) is twice the other root, what is the value of (k)?HardLevel 33If (\alpha,\beta) are the roots of (2x^2-7x+3=0), what is the value of ((\alpha-\beta)^2)?HardLevel 33If the quadratic equation \(x^2+2(k-1)x+k+5=0\) has equal roots, what are the possible values of \(k\)?HardLevel 33If (\alpha,\beta) are the roots of (x^2-6x+m=0) and (\alpha^3+\beta^3=72), what is (m)?HardLevel 33If the difference between the roots of (x^2+(a-2)x+a=0) is (3), what is the value of (a)?HardLevel 33The roots of x^2-5x+6=0 are α and β. Which equation has roots α+1 and β+1?HardLevel 33If the roots of (3x^2+px+12=0) are in the ratio (1:4), what are the possible values of (p)?HardLevel 33When will the real roots of the quadratic equation \(ax^2+bx+c=0\), where \(a\ne0\), be additive inverses of each other?HardLevel 33For (kx^2+6x+9=0) to have real roots with (k\ne0), which condition on (k) is correct?HardLevel 33If the two roots of the quadratic equation \(x^2+bx+c=0\) are opposites of each other, which of the following conditions is necessary?HardLevel 33If (\alpha,\beta) are the roots of (x^2+4x+1=0), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?HardLevel 33Which statement is correct about the roots of the equation \(5x^2-2x+1=0\)?HardLevel 33If \(x=2\) is a root of the equation \(kx^2-6x+4=0\), what is the value of \(k\)?HardLevel 33Can the sum of the roots of ((m+1)x^2-2(m-1)x+(m-3)=0) be (2)?HardLevel 33If (\alpha,\beta) are the roots of (x^2-3x-2=0), which equation has roots (\alpha^2,\beta^2)?HardLevel 33If the roots of a quadratic equation are (2+\sqrt{5}) and (2-\sqrt{5}), which is the equation?HardLevel 33