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 equation \(x^2+ax+b=0\) are \(2+\sqrt{3}\) and \(2-\sqrt{3}\), what is the value of \(a+b\)?ExpertLevel 31If (\alpha,\beta) are the roots of (x^2-3x-1=0), what is (\alpha^2+3\beta+\alpha\beta) when (\alpha) is the larger root?ExpertLevel 31If (\alpha,\beta) are the roots of (x^2-3x-1=0), what is the correct value of (\alpha^2+3\beta+\alpha\beta)?ExpertLevel 31Which statement is correct about the roots of x^2-2px+p^2-9=0?HardLevel 31If both roots of (x^2+4x+c=0) are real and negative, which condition is necessary and sufficient?ExpertLevel 31If (\alpha,\beta) are the roots of (x^2-2x-8=0), what is ((\alpha+3)(\beta+3))?ExpertLevel 31If the roots of (2x^2-(3t+1)x+t^2+t=0) are (t) and (\frac{t+1}{2}), for which values is this statement true?ExpertLevel 31If one root of (x^2-(m-2)x+m-6=0) is (3), what is the other root?ExpertLevel 31If one root of (x^2-(m+2)x+3m=0) is (3), what is the other root?ExpertLevel 31What is the correct conclusion about the roots of (x^2-(2a+1)x+a(a+1)=0)?HardLevel 31If (\alpha,\beta) are the roots of (x^2-9x+18=0), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?ExpertLevel 31If the roots of (x^2-4x+k=0) are (\sin \theta) and (\cos \theta), what is the maximum possible value of (k)?ExpertLevel 31If the roots of the quadratic equation \(x^2-2x+k=0\) are \(\tan\theta\) and \(\cot\theta\), what is the value of \(k\)?ExpertLevel 31If the roots of \(x^2+ax+b=0\) are \(\frac{1}{2+\sqrt{3}}\) and \(\frac{1}{2-\sqrt{3}}\), what is the value of \(a\)?ExpertLevel 31If the product of the roots of the equation \(x^2-2(a+1)x+a^2-1=0\) is zero, which values of \(a\) are possible?ExpertLevel 31If (alpha,beta) are the roots of (x^2-6x+n=0) and (alpha^2+beta^2=20), what is the value of (n)?ExpertLevel 31If the two roots of the quadratic equation \(ax^2+bx+c=0\), where \(a\ne0\), are reciprocals of each other, which relation among the coefficients is necessary?ExpertLevel 31If the roots of (x^2-12x+q=0) are in the ratio (2:3), what is the value of (q)?ExpertLevel 31What are the roots of the quadratic equation \(x^2-(a+b)x+ab=0\)?ExpertLevel 31If (\alpha,\beta) are roots of (x^2+5x+r=0) and (\alpha^2\beta+\alpha\beta^2=-20), what is (r)?ExpertLevel 31