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 (\alpha,\beta) are roots of (2x^2-7x+3=0), what is the positive value of (\alpha-\beta)?If the roots of the equation \(x^2-(2a+3)x+(a^2+3a+2)=0\) are consecutive integers, what are the roots?For (x^2+2x+\lambda=0) to have real distinct roots and both negative roots, what is the correct condition on (\lambda)?If (\alpha,\beta) are roots of (x^2-5x+6=0), what is (\alpha^2-4\alpha+\beta^2-4\beta)?If (\alpha,\beta) are roots of (x^2-5x+6=0), what is the correct value of (\alpha^2-4\alpha+\beta^2-4\beta)?If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+6x+8=0\), what is the value of \((\alpha-\beta)^2+2\alpha\beta\)?If the roots of the quadratic equation \(x^2+px+q=0\) are \(-2\) and \(5\), what is the value of \(p-q\)?If one root of (2x^2-(h+1)x+h=0) is always (\frac{1}{2}), what is the other root?Assume that \(a\) is a real number. What is the nature of the roots of the equation \(x^2-2x+(a^2+1)=0\)?If (\alpha,\beta) are roots of (x^2-4x+c=0) and (\alpha^2+\beta^2=10), what are the roots of the equation?If (x^2+bx+25=0) has equal and negative roots, what is the value of (b)?If (\alpha,\beta) are roots of (x^2+x-1=0), what is (\alpha^5+\beta^5)?If the two roots of the quadratic equation \(x^2-(u+v)x+uv=0\) are equal, which relation between \(u\) and \(v\) is correct?If (\alpha,\beta) are the roots of (x^2-(2r+5)x+(r^2+5r+6)=0), what is the positive value of (\alpha-\beta)?If the positive difference between the roots of (4x^2-12x+m=0) is (1), what is the value of (m)?If the quadratic equation \(x^2-2(a-4)x+a^2-20=0\) has equal roots, what is the value of \(a\)?If (\alpha,\beta) are the roots of (x^2-9x+14=0), which equation has roots (\alpha-3) and (\beta-3)?Which statement is always true for (x^2-(a+6)x+6a=0)?If (\alpha,\beta) are the roots of (x^2-6x+3=0), what is (\frac{1}{\alpha^2}+\frac{1}{\beta^2})?If (kx^2-10x+k=0) has real reciprocal roots, which condition on (k) is correct?