Recent questions tagged quadratic-equations

A quadratic equation $x^{2}+b x+c=0$ has two real roots. If the difference between the reciprocals of the roots is $\frac{1}{3}$, and the sum of the reciprocals of the sq...

Suppose one of the roots of the equation $ax^{2} – bx + c = 0$ is $2 + \sqrt{3},$ where $a, b$ and $c$ are rational numbers and $a \neq 0.$ If $b = c^{3}$ then $|a|$ equa...

A person buys tea of three different qualities at ₹$ \; 800,$ ₹ $\; 500,$ and ₹$ \; 300 \; \text{per kg},$ respectively, and the amounts bought are in the proportion $2:3...

Let $\text{m}$ and $\text{n}$ be positive integers, If $x^{2} + mx + 2n = 0$ and $x^{2} + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is $7$...

If $x= \frac{\sqrt{p^{2}+q^{2}}+\sqrt{p^{2}-q^{2}}}{{\sqrt{p^{2}+q^{2}}-\sqrt{p^{2}-q^{2}}}}$ then $q^{2}x^{2}-2p^{2}x+q^{2}$ equals to :$3$$-1$$-2$$0$

sum of roots of the equation $\dfrac{3x^{3}-x^{2}+x-1}{3x^{3}-x^{2}-x+1}=\dfrac{4x^{3}-7x^{2}+x+1}{4x^{3}+7x^{2}-x-1}$ is :$0$$1$$-1$$2$

The quadratic equation $x^{2}+bx+c=0$ has two roots $4a$ and $3a$, where a is an integer. Which of the following is a possible value of $b^{2}+c$?$3721$$549$$427$$361$

Let $A$ be a real number. Then the roots of the equation $x^{2}-4x-\log _{2}A=0$ are real and distinct if and only if$A \frac{1}{16}$$A \frac{1}{8}$$A< \frac{1}{16}$$A< \...

If $a$ and $b$ are integers such that $2x^{2}- ax + 2 0$ and $x^{2}-bx+8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a-6b$ is _________

The minimum possible value of the sum of the squares of the roots of the equation $x^{2}+\left ( a+3 \right )x-\left ( a+5 \right )=0$ is$1$$2$$3$$4$

If $f_{1}\left ( x \right )=x^{2}+11x+n$ and $f_{2}\left ( x \right )=x$, then the largest positive integer $n$ for which the equation $f_{1}\left ( x \right )=f_{2}\left...

Given the quadratic equation $x^2 – (\text{A} – 3)x – (\text{A} – 2)$, for what value of $\text{A}$ will the sum of the squares of the roots be zero?$-2$$3$$6$$\text{None...

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2}-10x+15= 0$, then find the quadratic equation whose roots are $\bigg(\alpha+\dfrac{\alpha }{\beta }...

If the roots of the equation $(a^{2}+b^{2})x^{2}+2(b^{2}+c^{2})x+(b^{2}+c^{2})= 0$ are real, which of the following must hold true?$c^{2}\geq a^{2}$$c^{4}\geq a^{2}(b^{2}...

For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive?$x^{2}-y^{2}=0$$(x-k)^{2}+y^{2}=1$$2$$0$$\s...

Let $f(x) = ax^2 + bx +c$, where $a, b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2)$ and that $3$ is a root of $f(x)=0$.What is the val...

For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive?$x^2 - y^2 =0$$(x-k)^2 + y^2 =1$$2$$0$$\sqr...