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Quadratic Equations — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsQuadratic Equations4 Marks
Question
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide.
Image
Based on the above information, answer the following questions:
(i) Write an algebraic equation depicting the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) What should be the new dimensions of the enlarged photo?
(iv) Can any rational value of $x$ make the new area equal to $220 cm^2$ ?
✓

Answer

(i) Original area of photo $=18 \times 12 cm^2=216 cm^2$
Area of photo by increasing length and breadth each by $x$ units.
$
=(18+x)(12+x)=216+30 x+x^2
$
It is given that the area of photo is doubled by increasing length and width each by $x$ uints.
$
\therefore \quad 216+30 x+x^2=2 \times 216
$
(ii) Simplifying equation (i), we obtain $x^2+30 x-216=0$.
This is the required quadratic equation in standard form.
$
\begin{array}{ll}
\text { (iii) } & x^2+30 x-216=0 \\
\Rightarrow & x^2+36 x-6 x-216=0 \\
\Rightarrow & x(x+36)-6(x+36)=0 \Rightarrow(x-6)(x+36)=0 \Rightarrow x-6=0 \Rightarrow x=6 \quad[\because x+36 \neq 0]
\end{array}
$
Hence, the new dimensions of the enlarged photo are
$
\text { Length }=18+x=(18+6) cm=24 cm, \text { Breadth }=12+x=(12+6) cm=18 cm
$
(iv) If new area is $220 cm^2$, then
$
\begin{array}{ll}
& 216+30 x+x^2=220 \Rightarrow x^2+30 x-4=0 \Rightarrow x=-\frac{-30 \pm \sqrt{916}}{2}=-15 \pm \sqrt{229} \quad[\because x>0] \\
\Rightarrow & x=-15 \pm \sqrt{229}
\end{array}
$
Clearly, $x$ is not rational. Hence, no rational value of $x$ can make new area equal to $220 cm^2$.

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