Embibe Experts Solutions for Chapter: Quadratic Equations, Exercise 2: Exercise-2

The equation, ${\pi }^x=-2x^2+6x-9$ has

Let $f\left(x\right)=a{x}^2+bx+c>0, \forall x\in R$ or $f\left(x\right)<0, \forall x\in R$. Which of the following is/are CORRECT?

Let $f\left(x\right)=\frac{3}{x-2}+\frac{4}{x-3}+\frac{5}{x-4}$, then $f\left(x\right)=0$ has

If the quadratic equations $a{x}^2+bx+c=0 \left(a,b,c\in R, a\ne 0\right)$ and $x^2+4x+5=0$ have a common root, then $a,b,c$ must satisfy the relations

If the quadratic equations $x^2+abx+c=0$ and $x^2+acx+b=0$ have a common root, then the equation containing their other roots is/are

Consider the following statements

${\mathbf{S}}_1:$ The equation $2x^2+3x+1=0$ has irrational roots.

${\mathbf{S}}_2:$ If $a<b<c<d,$ then, the roots of the equation $\left(x-a\right)\left(x-c\right)+2\left(x-b\right)\left(x-d\right)=0$ are real and distinct.

${\mathbf{S}}_3:$ If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have a common root and $a,b,c\in N$, then the minimum value of $\left(a+b+c\right)$ is $10$.

${\mathbf{S}}_4:$ The value of the biquadratic expression $x^4-8x^3+18x^2-8x+2,$ when $x=2+\sqrt{3}$, is $1$.

Which of the following are CORRECT?

If the equations $x^2+ax+12=0, x^2+bx+15=0$& $x^2+\left(a+b\right)x+36=0$ have a common positive root, then which of the following are true?

If $x^2+\lambda x+1=0, \lambda \in \left(-2,2\right)$ and $4x^3+3x+2c=0$ have common roots, then $c+\lambda$ can be