List of top Mathematics Questions on Quadratic Equations

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:

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 \[ a\beta-a+\frac{\gamma}{\delta}= \]

The roots of the quadratic equation $x^2 + 5x + 6 = 0$ will be :

If the quadratic equations \[ x^2+2x-4k=0 \] and \[ x^2+9x+3k=0 \] have exactly one common root and \(k\neq 0\), then the equation having \(k\) as a root is

The number of real solution of equation x$|$x+4$|$+3$|$x+2$|$+10=0 is/are :

Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

By selling an article for Rs. 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.

Venkat can row a boat in still water at the speed of $12\text{ km/h}$. He ferries tourists $15\text{ km}$ upstream and $18\text{ km}$ downstream in $3$ hours. Find the speed of the stream.

If \( \alpha, \beta, \gamma, \delta \) are the roots of the equation \( x^4 - 10x^3 + 35x^2 - 50x + 24 = 0 \) and \( \alpha^2, \beta^2, \gamma^2, \delta^2 \) are the roots of the equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \), then \( a = \)

If two roots of the equation \( x^5 - 9x^4 + 27x^3 - 23x^2 - 24x + 36 = 0 \) are repeated roots of multiplicity 2 and all the roots are integers then the sum of the squares of all different roots of the equation is:

If \( \alpha, \beta \in \mathbb{R} \) and \( \alpha \leq \frac{x^2 - 3x + 4}{x^2 - 4x + 5} \leq \beta \) for all \( x \in \mathbb{R} \), then \( (2\alpha - 3)^2 + (2\beta - 3)^2 = \)

If the roots of the quadratic equation \(5x^2 - 10x + k = 0\) are real and equal, then the value of k is :

In an auditorium, the sloping floor allows the seats to be arranged to give a clear view of the stage. The seats are arranged in such a way that the number of rows is equal to the number of seats in each row. When the number of rows are doubled and the number of seats in each row is reduced by 16, then the total number of seats increases by 320. Based on the above information, the total number of seats in the original arrangement is :

If the minimum value of \[ f(x)=x^2+2bx+2c^2 \] is greater than the maximum value of \[ g(x)=-x^2-2cx+b^2, \] \(x\) being real, then

The roots of the quadratic equation \(4x^2 - (a - 1)^2 = 0\) are :

Find two consecutive negative integers, sum of whose squares is 481.

The roots of the quadratic equation $(x - 1)^2 = 16$ are :

If \(x = \sqrt{3}\) is a solution of the equation \(ax^2 + \sqrt{3}x - 12 = 0\), then :

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=

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: