Let's go through the solution to this problem step-by-step.
### (a) Number of roots of the quadratic equation
Given that the length of a rectangular plot is 8 meters more than its breadth and the area is 384 square meters, we can set up the following equation. Let the breadth be [tex]\( x \)[/tex] meters.
Then the length will be [tex]\( x + 8 \)[/tex] meters.
The area of the rectangle is given by:
[tex]\[
\text{Area} = \text{length} \times \text{breadth}
\][/tex]
[tex]\[
384 = (x + 8) \times x
\][/tex]
This expands to:
[tex]\[
384 = x^2 + 8x
\][/tex]
We rearrange this to form a standard quadratic equation:
[tex]\[
x^2 + 8x - 384 = 0
\][/tex]
In general, a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] has exactly 2 roots. Therefore, the number of roots is:
[tex]\[
\boxed{2}
\][/tex]
### (b) Length and breadth of the plot
Next, we need to solve the quadratic equation to find [tex]\( x \)[/tex]:
Equation:
[tex]\[
x^2 + 8x - 384 = 0
\][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
Here [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = -384 \)[/tex].
Calculating the discriminant:
[tex]\[
b^2 - 4ac = 8^2 - 4 \times 1 \times (-384) = 64 + 1536 = 1600
\][/tex]
The square root of the discriminant is:
[tex]\[
\sqrt{1600} = 40
\][/tex]
So, the roots are:
[tex]\[
x = \frac{-8 \pm 40}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x_1 = \frac{-8 + 40}{2} = \frac{32}{2} = 16
\][/tex]
[tex]\[
x_2 = \frac{-8 - 40}{2} = \frac{-48}{2} = -24
\][/tex]
Since dimensions cannot be negative, we take the positive value:
[tex]\[
x = 16
\][/tex]
So, the breadth of the plot [tex]\( x \)[/tex] is:
[tex]\[
\text{Breadth} = 16 \, \text{m}
\][/tex]
And the length, which is [tex]\( x + 8 \)[/tex], is:
[tex]\[
\text{Length} = 16 + 8 = 24 \, \text{m}
\][/tex]
Thus, the dimensions of the plot are:
[tex]\[
\boxed{\text{Length} = 24 \, \text{m}, \text{Breadth} = 16 \, \text{m}}
\][/tex]
### (c) Decrease in length to form a square plot
To convert the rectangular plot into a square, the length should be equal to the breadth. Given that the breadth is 16 meters, the length should also be 16 meters.
Currently, the length is 24 meters. Therefore, the length needs to be decreased by:
[tex]\[
\text{Decrease in length} = 24 - 16 = 8 \, \text{meters}
\][/tex]
So, the length should be decreased by:
[tex]\[
\boxed{8 \, \text{meters}}
\][/tex]
