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
What is the nature of the roots of the equation \(x^2-2kx+k^2=0\)?HardLevel 31If 2 and 3 are the roots of the equation \(ax^2+bx+6=0\), what is the value of \(a\)?HardLevel 31If the roots of the quadratic equation x² − (m + 2)x + 12 = 0 are 3 and 4, what is the value of m?HardLevel 31If the roots of the quadratic equation \(x^2+px+q=0\) are \(-2\) and \(-5\), what is the value of \(p+q\)?HardLevel 31If the roots of the quadratic equation \(x^2+6x+k=0\) are \(-2\) and \(-4\), what is the value of \(k\)?HardLevel 31What is the sum of reciprocals of the roots of (2x^2-7x+3=0)?HardLevel 31If (\alpha) and (\beta) are roots of (x^2-9x+20=0), what is the value of (\alpha^3+\beta^3)?HardLevel 31If the roots of the quadratic equation \(x^2-4x+k=0\) are real and equal, what is the value of \(k\)?HardLevel 31If the quadratic equation \(kx^2-6x+9=0\) has equal roots and \(k\ne0\), what is the value of \(k\)?HardLevel 31How many real roots does the equation \(x^2+2x+5=0\) have?HardLevel 31If the roots of x² − (a + 1)x + a = 0 are 1 and a, why is this correct?HardLevel 31If (\alpha) and (\beta) are roots of (x^2-6x+8=0), what is the value of ((\alpha+1)(\beta+1))?HardLevel 31If (\alpha) and (\beta) are roots of (x^2-3x-10=0), what is the value of (\alpha^2\beta+\alpha\beta^2)?HardLevel 31If (\alpha) and (\beta) are roots of (2x^2-5x+2=0), what is the value of (\frac{1}{\alpha^2}+\frac{1}{\beta^2})?HardLevel 31If the roots of (x^2+px+16=0) are equal and both negative, what is the value of (p)?HardLevel 31If one root of (x^2+ax+a=0) is (1), what will be the other root?HardLevel 31If the difference of roots of (x^2-5x+q=0) is (1), what is the value of (q)?HardLevel 31If (\alpha) and (\beta) are roots of (x^2-Sx+P=0), what is (\alpha^2+\beta^2) equal to?HardLevel 31If each root of (x^2-4x-12=0) is increased by (1), which monic equation is formed from the new roots?HardLevel 31If 1 is subtracted from each root of the equation \(x^2-4x-12=0\), what is the product of the resulting roots?HardLevel 31