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 roots of the quadratic equation \(x^2+ax+b=0\) are \(4+\sqrt{7}\) and \(4-\sqrt{7}\), what is the value of \(a+b\)?ExpertLevel 33If (\alpha,\beta) are roots of (x^2-5x-3=0), what is (\alpha^2+5\beta+\alpha\beta)?ExpertLevel 33What is the positive difference between the two roots of the quadratic equation \(x^2-2tx+t^2-49=0\)?ExpertLevel 33If both roots of (x^2-6x+c=0) are real and positive, what is the correct condition on (c)?ExpertLevel 33If (\alpha,\beta) are roots of (x^2-4x-12=0), what is ((\alpha-5)(\beta-5))?ExpertLevel 33If the roots of (4x^2-(5t+3)x+t(t+3)=0) are said to be (t) and (\frac{t+3}{4}), when is this statement true?ExpertLevel 33If one root of \(x^2-(m+9)x+9m=0\) is \(9\), what is the other root?ExpertLevel 33What are the roots of the equation \(x^2-2(a+2)x+a^2+4a=0\)?ExpertLevel 33If (\alpha,\beta) are the roots of (x^2-13x+36=0), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?ExpertLevel 33If the roots of (x^2-sx+1=0) are (2\tan\theta) and (\frac{1}{2}\cot\theta), what is the correct reality condition on (s)?ExpertLevel 33If the roots of (x^2+ax+b=0) are (\frac{1}{4+\sqrt{3}}) and (\frac{1}{4-\sqrt{3}}), what is the value of (a)?ExpertLevel 33If the product of the roots of the equation \(x^2-2(a-3)x+a^2-16=0\) is zero, what are the values of \(a\)?ExpertLevel 33For the quadratic equation \(ax^2+bx+c=0\), where \(a\neq0\) and \(c\neq0\), which condition is necessary and sufficient for its two roots to be reciprocals of each other?ExpertLevel 33If the roots of 7x² − 6x + λ = 0 are not real, what is the correct condition on λ?MediumLevel 33If the roots of the equation \(x^2-16x+q=0\) are in the ratio \(5:3\), what is the value of \(q\)?ExpertLevel 33What are the roots of the equation \(x^2-(u+2v)x+2uv=0\)?ExpertLevel 33If (\alpha,\beta) are roots of (x^2-7x+r=0) and (\alpha^2\beta+\alpha\beta^2=84), what is (r)?ExpertLevel 33If (\alpha,\beta) are roots of (x^2-5x+1=0), what is (\alpha^4+\beta^4)?ExpertLevel 33If both roots of (x^2-14x+m=0) are prime numbers, what is the sum of possible values of (m)?ExpertLevel 33If (x^2+px+64=0) has equal and positive roots, what is the value of (p)?ExpertLevel 33