NCERT Solutions for Chapter: Quadratic Equations, Exercise 4: Exercise 4.4

Represent the following situation mathematically:
A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be $55$ minus the number of toys produced in a day. On a particular day, the total cost of production was $₹750$. We would like to find out the number of toys produced on that day.

Check whether the following is a quadratic equation:

${\left(x-2\right)}^2+1=2x-3$

Check whether the following is a quadratic equation:
 ${\left(x+2\right)}^3=x^3-4$

Check whether the following is a quadratic equation:
$x\left(2 x+3\right)=x^2+1$
 

Find the roots of the following quadratic equations, if they exist, using the quadratic formula:

 $x^2+4x+5=0$ 

Find the roots of the following quadratic equations, if they exist, using the quadratic formula:

 $2x^2-2\sqrt{2}x+1=0$ 

A pole has to be erected at a point on the boundary of a circular park of diameter $13$ metres in such a way that the differences of its distances from two diametrically opposite fixed gates $A$ and $B$ on the boundary is $7$ metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Find the discriminant of the equation $3x^2-2x+\frac{1}{3}=0$ and hence find the nature of its roots. Find them, if they are real.