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 α and β are roots of x² − 3x + 2 = 0, which equation has 2α and 2β as roots?HardLevel 31If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-5x+4=0\), what is the sum of \((\alpha+2)\) and \((\beta+2)\)?HardLevel 31If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-10x+25=0\), what is the value of \(\alpha-\beta\)?HardLevel 31For the quadratic equation \(ax^2+bx+c=0\) with integer coefficients, where \(a\ne0\), which condition ensures that both roots are distinct real irrational numbers? Here, the discriminant is \(\Delta=b^2-4ac\).HardLevel 31If the sum of roots is (4) and the sum of their squares is (20), what is the product of roots?HardLevel 31If the roots of (x^2-6x+c=0) are in the ratio (1:2), what is the value of (c)?HardLevel 31If the roots of x² + px + 12 = 0 are in the ratio 1:3 and both are negative, what is p?HardLevel 31If the two roots of the quadratic equation \(x^2+bx+9=0\) are equal and positive, what is the value of \(b\)?HardLevel 31If the product of roots of (3x^2-2x+k=0) equals the sum of roots, what is (k)?HardLevel 31If the equation \(x^2-2x+(m+1)=0\) has no real roots, which condition on \(m\) is correct?HardLevel 31If the two roots of the quadratic equation \(x^2-2(m+1)x+(m^2+2m)=0\) are equal, what is the value of \(m\)?HardLevel 31If roots of (x^2-2(m+1)x+(m^2+2m+1)=0) are equal, what is true for (m)?HardLevel 31If calphac and cbetac are the roots of cx^2-7x+10=0c, what is the value of (alpha-1)(beta-1)c?HardLevel 31If 2 and −3 are the roots of the equation \(3x^2+px+q=0\), what is the value of \(\frac{q}{3}\)?HardLevel 31If the roots of the quadratic equation \(x^2+px+q=0\) are \(4\) and \(-1\), what is the value of \(p-q\)?HardLevel 31If alpha and beta are the roots of the equation x^2-4x+1=0, what is the value of alpha^2+beta^2?HardLevel 31If (\alpha) and (\beta) are roots of (x^2+x-1=0), what is the value of (\alpha^2+\alpha\beta+\beta^2)?HardLevel 31Which pair gives the roots of the equation \(x^2-(2m+1)x+m(m+1)=0\)?HardLevel 31If (\alpha) and (\beta) are roots of (x^2-8x+15=0), what is the value of (\frac{\alpha-1}{\beta-1}+\frac{\beta-1}{\alpha-1})?HardLevel 31Which of the following quadratic equations has real and distinct roots that are irrational?HardLevel 31