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Question

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the bill each person had to pay, would have reduced by Rs. 200. Find the number of people

staying overnight.

(A) 5

(B) 8

(C) 6

(D) 7

Answer
Verified

Quadratic equations- These are the Polynomial equations of degree 2 in one variable.

The quadratic equation will always have two roots. The nature of roots may be real or imaginary.

Lets understand them with and example-:

x² + 5x + 6 = 0

There are two roots of this equation. i.e. -2 and -3

When we place \[x = - 2\] or \[x = - 3\]

In the above equation they will satisfy it.

So, roots of equation- Real number will be called a solution or a root if it satisfies the equation.

I.e. a will be the root of equation,

If \[f\left( a \right) = 0\]

The solution or roots of a quadratic equation are given by the quadratic formula:

For x people the bill will be $ = $ Rs. 48000/.

So, for 1 person bill will be $ = Rs\left( {\dfrac{{4800}}{x}} \right)$

According to the question :

When 4 persons will more, bill will be $ = Rs\left( {\dfrac{{4800}}{{x + 4}}} \right)$

After adding 4 persons, bill paid by each person will reached by 200/-

Therefore

$\left( {\dfrac{{4800}}{x}} \right) - \left( {\dfrac{{4800}}{{x + 4}}} \right) = 200$

$4800(x + 4) - 4800x = x(x + 4) \times 200$

$4800x + 19200 - 4800x = ({x^2} + 4x)200$

$19200 = ({x^2} + 4x) \times 200$

$96 = {x^2} + 4x$ or ${x^2} + 4x - 96 = 0$

Which is a quadratic equation

So, we will calculate the roots of this equation by quadratic equation.

We know that $x = \dfrac{{ - B \pm \sqrt {{B^2} - 4AC} }}{{2A}}$

Where A $ = $ coefficient of ${x^2} = 1$

B $ = $ coefficient of $x = 4$

C $ = $ coefficient of $ = - 96$

Therefore,

$x = \dfrac{{ - 4 \pm \sqrt {{{(4)}^2} - 4(1)( - 96)} }}{{2 \times 1}}$

$x = \dfrac{{ - 4 \pm \sqrt {16 + 384} }}{2}$

$x = \dfrac{{ - 4 \pm \sqrt {400} }}{2}$

$x = \dfrac{{ - 4 \pm 20}}{2}$

$x = \dfrac{{ - 4 + 20}}{2}$ or $x = \dfrac{{ - 4 - 20}}{2}$

$x = 8$ or $x = - 12$

Since x denotes number of people,

So it can't be negative.

Therefore we will neglect the negative digit.

Hence,

The number of people staying over night \[ = x = 8.\]

Therefore option B i.e. 8 is the correct option.

\[x = \left( { - b \pm \surd D} \right)/2a\]

where,

; and called discriminant of equation.

We can determine the nature of roots by using this Discriminant.

(¡) \[D > 0,\] roots are real and distinct (unequal)

(¡¡)\[D = 0,\] roots are real and equal

(¡¡¡) \[D < 0,\] roots are imaginary and unequal