Recent questions tagged quadratic-equations

The product of two numbers is $120,$ and their sum is $22.$ What are the numbers?$12,11$$6,20$$8,15$$10,12$ 

Consider the equation $x^{7} + 4x^5 + 2x^3 + x + 1 = 0.$ The number of real roots of this equation is:(A) 1 (B) 3 (C) 5 (D) 7

If the difference between the roots of the equation \( x^2 + bx + 4 = 0 \) is less than \( \sqrt{48} \), then the set of possible values of \( b \) is ________\((-10, 10)...

Find all possible integers $n$ for which $n^{2}+20 n+15$ is a perfect square.

What is the radix of the numbers if the solution to the quadratic equation $x^2-10x+26=0$ is $x=4$ and $x=7$?$8$$9$$10$$11$

Let $x_1$ and $x_2$ be the roots of the quadratic equation $x^2-3x+a=0$, and $x_3$ and $x_4$ be the roots of the quadratic equation $x^2-12x+b=0$. If $x_1, x_2, x_3$ and ...

The conditions on $a$, $b$ and $c$ under which the roots of the quadratic equation $ax^2+bx+c=0 \: ,a \neq 0, \: b \neq 0 $ and $c \neq 0$, are unequal magnitude but of t...

Consider the equation $P(x) =x^3+px^2+qx+r=0$ where $p,q$ and $r$ are all real and positive. State which of the following statements is always correct.All roots of $P(x) ...

If $x$ is real, the set of real values of $a$ for which the function $$y=x^2-ax+1-2a^2$$ is always greater than zero is$- \frac{2}{3} < a \leq \frac{2}{3}$$- \frac{2}{3} ...

If $a,b,c$ are sides of a triangle $ABC$ such that $x^2-2(a+b+c)x+3 \lambda (ab+bc+ca)=0$ has real roots then$\lambda < \frac{4}{3}$$\lambda \frac{5}{3}$$\lambda \in \bi...

The number of real roots of the equation$$2 \cos \left( \frac{x^2+x}{6} \right) = 2^x +2^{-x} \text{ is }$$$0$$1$$2$infinitely many

The number of real solutions of the equations $(9/10)^x = -3+x-x^2$ is$2$$0$$1$none of the above

If two real polynomials $f(x)$ and $g(x)$ of degrees $m\: (\geq 2)$ and $n\: (\geq 1)$ respectively, satisfy$$f(x^2+1)=f(x)g(x),$$for every $x \in \mathbb{R}$, then$f$ ha...

Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is$2$$4$$3$$1$

If $\alpha$ and $\beta$ be the roots of the equation $x^2+3x+4=0$, then the equation with roots $(\alpha + \beta)^2$ and $(\alpha – \beta)^2$ is$x^2+2x+63=0$$x^2-63x+2=0$...

If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...

If one root of a quadratic equation $ax^2+bx+c=0$ be equal to the $n^{th}$ power of the other, then$(ac)^{\frac{n}{n+1}} +b=0$$(ac)^{\frac{n+1}{n}} +b=0$$(ac^{n})^{\frac{...

Let $p,q,r,s$ be real numbers such that $pr=2(q+s)$. Consider the equations $x^2+px+q=0$ and $x^2+rx+s=0$. Thenat least one of the equations has real rootsboth these equa...

Let $x^{2}-2(4k-1)x+15k^{2}-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is$2$$4$$3$$1$

If $\alpha$ and $\beta$ be the roots of the equation $x^{2}+3x+4=0,$ then the equation with roots $(\alpha+\beta)^{2}$ and $(\alpha-\beta)^{2}$ is$x^{2}+2x+63=0$$x^{2}-63...

If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...

If one root of a quadratic equation $ax^{2}+bx+c=0$ be equal to the n th power of the other, then$(ac)^{\frac{n}{n+1}}+b=0$$(ac)^{\frac{n+1}{n}}+b=0$$(ac^{n})^{\frac{1}{n...