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Quadratic Equations in One Variable — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsQuadratic Equations in One Variable4 Marks
Question
Harish made a rectangular garden, with its length $5$ metres more than its width. The next year, he increased the length by $3$ metres and decreased the width by $2$ metres. If the area of the second garden was $119$ sq m, was the second garden larger or smaller ?
✓

Answer

In first case,
Let length of the garden $= x m$
then width $= (x – 5) m$
Area $= l x b = x(x – 5)$ sq. m
In second case,
Length $= (x + 3)m$
and width $= x - 5 - 2 = (x - 7)m$
According to the condition,
$(x + 3)(x - 7) = 119$
$\Rightarrow x^2 - 7x + 3x - 21 = 119$
$\Rightarrow x^2 - 4x - 21 - 119 = 0$
$\Rightarrow x^2- 4x - 140 = 0$
$\Rightarrow x^2 - 14x + 10x - 140 = 0$
$\Rightarrow x(x - 14) + 10(x - 14) = 0$
$\Rightarrow (x - 14)(x + 10) = 0$
Either $x - 14 = 0,$
then $x = 14$
or
$x + 10 = 0$,
then $x = -10,$
but it is not possible as it is negative.
$\therefore$ Length of first garden $= 14m$
and width $= 14 - 5 = 9m$
Area
$= l x b$
$= 14 x 9$
$= 126m^2$​​​​​​​
Difference of areas of two rectangles
$= 126 - 119$
$= 7sq.m.$
∴ Area of second garden is smaller than the area of the first garden by $7$ sq.m.

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