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
Which statement is always true for x^2-(a+6)x+6a=0?HardLevel 32If (\alpha,\beta) are the roots of (x^2-6x+3=0), what is (\frac{1}{\alpha^2}+\frac{1}{\beta^2})?ExpertLevel 32If (kx^2-10x+k=0) has real reciprocal roots, which condition on (k) is correct?ExpertLevel 32If the roots of (x^2+px+15=0) are (r) and (r+2), what are the possible values of (p)?ExpertLevel 32For the quadratic equation \(9x^2-6(a+1)x+a^2-3a=0\) to have real roots, which condition on \(a\) is necessary?ExpertLevel 32If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-7x+12=0\), what is the value of \(\frac{\alpha+1}{\alpha-1}+\frac{\beta+1}{\beta-1}\)?ExpertLevel 32If (\alpha,\beta) are the roots of (3x^2-8x+4=0), what is (\alpha^3+\beta^3)?ExpertLevel 32If the quadratic equation \(x^2+(m+1)x+16=0\) has equal roots, what are the possible values of \(m\)?ExpertLevel 32If the roots of the quadratic equation \(x^2+ax+b=0\) are \(3+\sqrt{2}\) and \(3-\sqrt{2}\), what is the value of \(a+b\)?ExpertLevel 32If (\alpha,\beta) are the roots of (x^2-4x-2=0), what is (\alpha^2+4\beta+\alpha\beta)?ExpertLevel 32What is the positive difference between the two roots of \(x^2-2qx+q^2-16=0\)?ExpertLevel 32If both roots of (x^2-5x+c=0) are real and positive, what is the correct condition on (c)?ExpertLevel 32If (\alpha,\beta) are the roots of (x^2-3x-10=0), what is ((\alpha-4)(\beta-4))?ExpertLevel 32If the roots of (3x^2-(4t+2)x+t(t+2)=0) are said to be (t) and (\frac{t+2}{3}), when is this statement true?ExpertLevel 32What are the roots of the quadratic equation x² − 2(a + 1)x + a² + 2a = 0?ExpertLevel 32If (\alpha,\beta) are the roots of (x^2-11x+24=0), what is (\frac{\alpha}{\beta}+\frac{\beta}{\alpha})?ExpertLevel 32If the roots of (x^2-sx+1=0) can be (\tan\theta) and (\cot\theta), what is the correct condition on (s) for real (\theta)?ExpertLevel 32If the roots of (x^2+ax+b=0) are (\frac{1}{3+\sqrt{5}}) and (\frac{1}{3-\sqrt{5}}), what is the value of (a)?ExpertLevel 32If the product of the roots of the equation \(x^2-2(a-2)x+a^2-9=0\) is zero, what are the values of \(a\)?ExpertLevel 32If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-8x+n=0\) and \(\alpha^2+\beta^2=40\), what is the value of \(n\)?ExpertLevel 32