Quadratic Equations

If $$ (3 + 2\sqrt{2}) $$ is a root of the equation $$ ax^2 + bx + c = 0 $$, and $$ (4 + 2\sqrt{3}) $$ is a root of the e...

The minimum possible value of $$\frac{x^2 - 6x + 10}{3-x}$$, for $$x < 3$$, is:...

Suppose $$k$$ is any integer such that the equation $$2x^2 + kx + 5 = 0$$ has no real roots, and the equation $$x^2 + (k...

Let $$f(x)$$ be a quadratic polynomial in $$x$$ such that $$f(x) \geq 0$$ for all real numbers $$x$$. If $$f(2) = 0$$ an...

The number of integer solutions of the equation $$(x^2 - 10)^{(x^2 - 3x - 10)} = 1$$ is:...

Let $$ a $$ and $$ b $$ be natural numbers. If $$ a^2 + ab + a = 14 $$ and $$ b^2 + ab + b = 28 $$, then $$ 2a ...

![Screenshot 2025-04-02 200731.png](https://res.cloudinary.com/aptidude/image/upload/v1743604663/aptidude_questions/hwls...

A quadratic equation $$ x^2 + bx + c = 0 $$ has two real roots. If the difference between the reciprocals of the roots ...

The equation $$ x^3 + (2r + 1)x^2 + (4r - 1)x + 2 = 0 $$ has $$ -2 $$ as one of the roots. If the other two roots a...

Let $$ \alpha $$ and $$ \beta $$ be the two distinct roots of the equation $$ 2x^2 - 6x + k = 0 $$, such that $$ (\...

If $$ x $$ and $$ y $$ are real numbers such that $$ x^2 + (x - 2y - 1)^2 = -4y(x + y), $$ then the value of $$ x -...

If $$x$$ and $$y$$ are real numbers such that $$4x^2 + 4y^2 - 4xy - 6y + 3 = 0,$$ then the value of $$(4x + 5y)$$ is ...

The roots $$\alpha$$, $$\beta$$ of the equation $$3x^2 + \lambda x - 1 = 0$$ , satisfy $$\frac{1}{\alpha^2} + \frac{1}...

If the equations $$x^2 + mx + 9 = 0$$, $$x^2 + nx + 17 = 0$$, and $$x^2 + (m+n)x + 35 = 0$$ have a common negative root,...

Let $$ k $$ be the largest integer such that the equation $$ (x - 1)^2 + 2kx + 11 = 0 $$ has no real roots. If $$ y ...

If $$(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$$, where $$a$$ and $$b$$ are natural numbers, then $$a + b$$ equals: A. 7 ...