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 difference between the roots of the equation \(x^2-(k+2)x+2k=0\) is 2, find the values of \(k\).For the equation \(x^2-2x+(p+3)=0\) to have no real roots, what is the correct condition on \(p\)?If (\alpha,\beta) are roots of (2x^2+3x-5=0), what is (\alpha^2\beta+\alpha\beta^2)?If the quadratic equation \(x^2+kx+16=0\) has equal roots, what are the possible values of \(k\)?If one root of (x^2+ax+12=0) is (3) more than the other root, what are the possible values of (a)?Which quadratic equation has roots (\frac{1}{2}) and (\frac{3}{4})?If α and β are roots of x^2+px+q=0 and α+1 and β+1 are roots of x^2-5x+6=0, what are p and q?For the equation \(x^2-2(k+1)x+k^2=0\) to have real and distinct roots, what is the correct condition on \(k\)?If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-5x+1=0\), what is the value of \(\alpha^2+\beta^2\)?If the roots of a quadratic equation are (3) and (-2), and the coefficient of (x^2) is (2), which is the equation?If (\alpha,\beta) are roots of (x^2+mx+n=0), (\alpha-\beta=4), and (\alpha\beta=5), what is (m^2)?Which statement is always true about x^2-(a+3)x+3a=0?If the roots of the quadratic equation \(x^2-10x+k=0\) are distinct prime numbers, what is the value of \(k\)?For (x^2+2x+c=0), the roots are real and their product is less than their sum. What is the correct condition on (c)?If the roots of the quadratic equation \(x^2-(m+1)x+m=0\) are reciprocals of each other, what is the value of \(m\)?If (2x^2+kx+2=0) has equal and negative roots, what is (k)?If (\alpha,\beta) are roots of (x^2-4x+2=0), what is ((\alpha+2)(\beta+2))?The difference between the two roots of the equation \(x^2+6x+r=0\) is \(2\sqrt{5}\). What is the value of \(r\)?If \(x=0\) is a root of the quadratic equation \(ax^2+bx+c=0\), where \(a\ne0\), which of the following conditions must be true?What are the two roots of the equation \(4x^2-4x+1=0\)?