List of top Mathematics Questions on Quadratic Equations

ABCD is a rectangle of dimensions 80 cm \(\times\) 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.

If all roots of \[ x^5-3x^4+2x^3-3x^2+5x-2=0 \] are increased by real value h so that term containing \(x^3\) vanishes in transformed equation and h is integer, then h=

Two real roots of \[ 3x^4+ax^3+55x^2-52x+12=0 \] are positive and equal. Product of other two roots is 1. If roots belong to natural numbers then \[ a\beta-a+\frac{\gamma}{\delta}= \]

If \(\alpha\in R\) and equation \[ (x-\alpha)(x-3)+1=0 \] has equal roots, then sum of squares of all values of \(\alpha\) is

If the equation \[ 2x^2-5x+k=0 \] has equal roots, then the value of \(k\) is

If \(\alpha,\beta\) are the roots of the quadratic equation \[ x^2-3x+1=0, \] then the value of \(\alpha^3+\beta^3\) is:

Let \(\alpha\) and \(\beta\) be the roots of \[ x^2+bx+c=0. \] If \[ \alpha^2+\beta^2=14 \] and \[ \alpha\beta=3, \] then the value of \(b^2\) is:

If the quadratic equation \((\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \lambda \neq -2\), has two positive roots, then the number of possible integral values of \(\lambda\) is:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - 3x + r = 0 \), and \( \frac{\alpha}{2}, 2\beta \) be the roots of the equation \( x^2 + 3x + r = 0 \). If the roots of the equation \( x^2 + 6x = m \) are \( 2\alpha + \beta + 2r \) and \( \alpha - 2\beta - \frac{r}{2} \), then \( m \) is equal to:

If \( \alpha,\beta \) where \( \alpha<\beta \), are the roots of the equation \[ \lambda x^2-(\lambda+3)x+3=0 \] such that \[ \frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, \] then the sum of all possible values of \( \lambda \) is:

Let \( \alpha, \alpha + 2 \in \mathbb{Z} \) be the roots of the quadratic equation} \[ x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \cdots + (x+n-1)(x+n+1) = 4n \] for some \( n \in \mathbb{N} \). Then \( n + \alpha \) is equal to:

Let $\alpha, \beta$ be the roots of the quadratic equation \[ 12x^2 - 20x + 3\lambda = 0,\ \lambda \in \mathbb{Z}. \] If \[ \frac{1}{2} \le |\beta-\alpha| \le \frac{3}{2}, \] then the sum of all possible values of $\lambda$ is

For what values of $k$ does the quadratic equation $9x^2 + kx + 1 = 0$ have equal roots?

What is the nature of the roots of the quadratic equation $x^2 - 7x + 10 = 0$?

Which of the following are roots of the quadratic equation $6x^2-x-2=0$?

The formula for finding the roots of quadratic equation is _____.

Verify that roots of the quadratic equation \((p - q)x^2 + (q - r)x + (r - p) = 0\) are equal when \(q + r = 2p\).

If the quadratic expression \[ (\lambda+2)x^2 - 3\lambda x + 4\lambda = 0, \qquad \lambda \ne -2 \] has two positive roots, then the number of possible integral values of \(\lambda\) is:

Let the mean and variance of 8 numbers -10, -7, -1, x, y, 9, 2, 16 be \( 2 \) and \( \frac{293}{4} \), respectively. Then the mean of 4 numbers x, y, x+y+1, |x-y| is:

The sum of all the roots of the equation \((x-1)^2 - 5|x-1| + 6 = 0\), is: