Nature of Roots

Find the roots of the quadratic equation by applying the quadratic formula.
 $2x^2+x+4=0$
 

Find the roots of the quadratic equation by applying the quadratic formula.
 $4x^2+4\sqrt{3}x+3=0$
 

Find the roots of the quadratic equation by applying the quadratic formula.
 $2x^2+x-4=0$

Find the values of $k$ for the following quadratic equation, so that it has two equal roots.

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them.

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them.

Is it possible to design a rectangular park of perimeter $80 \mathrm{m}$ and area $400 {\mathrm{m}}^2$? If so, find its length and breadth.

Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is $20$ years. Four years ago, the product of their ages in years was $48$.

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800 {\mathrm{m}}^2$? If so, find its length and breadth.

Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots.
 $2x^2+kx+3=0$

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them :
 $2x^2-3x+5=0$
 

Find the roots of the following equation.
 $x-\frac{1}{x}=3,x\ne 0$
 

Find the roots of the quadratic equation by applying the quadratic formula.
 $2x^2-7x+3=0$