Examine whether the following equation is quadratic or not: $(x-2)^{2}+1=2x-3$.

The faster train takes $1$ hour less than the slower train for covering a $400\, km$ distance. If the speed of the slower train is $20\, km/hr$ less than the speed of the faster train,find the speed of the slower train.

The golden ratio,often denoted by the Greek letter $\phi$,is a special number. Which of the following is the golden number?

If the roots of the following quadratic equation exist,find them by the method of completing the square: $3x - \frac{2}{x} - 7 = 0$.

Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $4x^{2}-6x+2=0$.

If the following quadratic equation has two equal and real roots,then find the value of $k$: $5x^{2} - 2kx + 20 = 0$.