Class 10: Quadratic Equations

$\displaystyle \textbf{Question 1: }\text{Ashu is }x\text{ years old while his mother Mrs Veena is }x^2\text{ years old. Five years}$
$\displaystyle \text{hence Mrs Veena will be three times old as Ashu. Find their present ages.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Ashu's present age}=x\text{ years.}$
$\displaystyle \therefore \text{Mrs Veena's present age}=x^2\text{ years.}$
$\displaystyle \text{Five years hence, Ashu's age}=(x+5)\text{ years.}$
$\displaystyle \text{Mrs Veena's age five years hence}=(x^2+5)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle x^2+5=3(x+5)$
$\displaystyle \Rightarrow x^2+5=3x+15$
$\displaystyle \Rightarrow x^2-3x-10=0$
$\displaystyle \Rightarrow (x-5)(x+2)=0$
$\displaystyle \Rightarrow x=5\text{ or }x=-2$
$\displaystyle \text{Since age cannot be negative, }x=-2\text{ is rejected.}$
$\displaystyle \therefore \text{Ashu's present age}=5\text{ years.}$
$\displaystyle \therefore \text{Mrs Veena's present age}=5^2=25\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{The sum of the ages of a man and his son is }45\text{ years. Five years ago, the}$
$\displaystyle \text{product of their ages was four times the man's age at the time. Find their present ages.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the present age of the man be }x\text{ years.}$
$\displaystyle \therefore \text{Present age of his son}=(45-x)\text{ years.}$
$\displaystyle \text{Five years ago, man's age}=(x-5)\text{ years.}$
$\displaystyle \text{Son's age five years ago}=(40-x)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle (x-5)(40-x)=4(x-5)$
$\displaystyle \Rightarrow (x-5)\{(40-x)-4\}=0$
$\displaystyle \Rightarrow (x-5)(36-x)=0$
$\displaystyle \Rightarrow x=5\text{ or }x=36$
$\displaystyle \text{If }x=5,\text{ the son's present age would be }40\text{ years, which is not possible.}$
$\displaystyle \therefore x=36$
$\displaystyle \therefore \text{Present age of the man}=36\text{ years.}$
$\displaystyle \therefore \text{Present age of his son}=45-36=9\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{The product of Shikha's age five years ago and her age }8\text{ years later is }30,\text{ her}$
$\displaystyle \text{age at both times being given in years. Find her present age.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let Shikha's present age be }x\text{ years.}$
$\displaystyle \therefore \text{Her age five years ago}=(x-5)\text{ years.}$
$\displaystyle \text{Her age }8\text{ years later}=(x+8)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle (x-5)(x+8)=30$
$\displaystyle \Rightarrow x^2+3x-40=30$
$\displaystyle \Rightarrow x^2+3x-70=0$
$\displaystyle \Rightarrow x^2+10x-7x-70=0$
$\displaystyle \Rightarrow x(x+10)-7(x+10)=0$
$\displaystyle \Rightarrow (x+10)(x-7)=0$
$\displaystyle \Rightarrow x=-10\text{ or }x=7$
$\displaystyle \text{Since age cannot be negative, }x=-10\text{ is rejected.}$
$\displaystyle \therefore \text{Shikha's present age is }7\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{The product of Ramu's age (in years) five years ago and his age (in years) nine}$
$\displaystyle \text{years later is }15.\text{ Determine Ramu's present age.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let Ramu's present age be }x\text{ years.}$
$\displaystyle \therefore \text{His age five years ago}=(x-5)\text{ years.}$
$\displaystyle \text{His age nine years later}=(x+9)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle (x-5)(x+9)=15$
$\displaystyle \Rightarrow x^2+4x-45=15$
$\displaystyle \Rightarrow x^2+4x-60=0$
$\displaystyle \Rightarrow x^2+10x-6x-60=0$
$\displaystyle \Rightarrow x(x+10)-6(x+10)=0$
$\displaystyle \Rightarrow (x+10)(x-6)=0$
$\displaystyle \Rightarrow x=-10\text{ or }x=6$
$\displaystyle \text{Since age cannot be negative, }x=-10\text{ is rejected.}$
$\displaystyle \therefore \text{Ramu's present age is }6\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Is the following situation possible? If so, determine their present ages.}$
$\displaystyle \text{The sum of the ages of two friends is }20\text{ years. Four years ago, the product of their}$
$\displaystyle \text{ages in years was }48.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the present age of one friend be }x\text{ years.}$
$\displaystyle \therefore \text{Present age of the other friend}=(20-x)\text{ years.}$
$\displaystyle \text{Four years ago, their ages were }(x-4)\text{ and }(16-x)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle (x-4)(16-x)=48$
$\displaystyle \Rightarrow -x^2+20x-64=48$
$\displaystyle \Rightarrow x^2-20x+112=0$
$\displaystyle \text{Here, }a=1,\ b=-20,\ c=112.$
$\displaystyle D=b^2-4ac=(-20)^2-4(1)(112)$
$\displaystyle =400-448=-48<0$
$\displaystyle \therefore \text{The equation has no real roots.}$
$\displaystyle \therefore \text{The given situation is not possible.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{A girl is twice as old as her sister. Four years hence, the product of their}$
$\displaystyle \text{ages (in years) will be }160.\text{ Find their present ages.}\hfill\text{[CBSE 2010]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the present age of the younger sister be }x\text{ years.}$
$\displaystyle \therefore \text{Present age of the girl}=2x\text{ years.}$
$\displaystyle \text{Four years hence, their ages will be }(x+4)\text{ and }(2x+4)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle (x+4)(2x+4)=160$
$\displaystyle \Rightarrow 2x^2+12x+16=160$
$\displaystyle \Rightarrow x^2+6x-72=0$
$\displaystyle \Rightarrow (x+12)(x-6)=0$
$\displaystyle \Rightarrow x=-12\text{ or }x=6$
$\displaystyle \text{Since age cannot be negative, }x=-12\text{ is rejected.}$
$\displaystyle \therefore \text{Present age of the younger sister}=6\text{ years.}$
$\displaystyle \therefore \text{Present age of the girl}=2(6)=12\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{The sum of the reciprocals of Rehman's ages (in years) }3\text{ years ago and }5\text{ years}$
$\displaystyle \text{from now is }\frac{1}{3}.\text{ Find his present age.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let Rehman's present age be }x\text{ years.}$
$\displaystyle \therefore \text{His age }3\text{ years ago}=(x-3)\text{ years.}$
$\displaystyle \text{His age }5\text{ years from now}=(x+5)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle \frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$
$\displaystyle \Rightarrow \frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}$
$\displaystyle \Rightarrow 3(2x+2)=(x-3)(x+5)$
$\displaystyle \Rightarrow 6x+6=x^2+2x-15$
$\displaystyle \Rightarrow x^2-4x-21=0$
$\displaystyle \Rightarrow (x-7)(x+3)=0$
$\displaystyle \Rightarrow x=7\text{ or }x=-3$
$\displaystyle \text{Since age cannot be negative, }x=-3\text{ is rejected.}$
$\displaystyle \therefore \text{Rehman's present age is }7\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{If Zeba were younger by }5\text{ years than what she really is, then the square of her age}$
$\displaystyle \text{(in years) would have been }11\text{ more than }5\text{ times her actual age. What is her age now?}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let Zeba's present age be }x\text{ years.}$
$\displaystyle \therefore \text{If she were }5\text{ years younger, her age would be }(x-5)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle (x-5)^2=5x+11$
$\displaystyle \Rightarrow x^2-10x+25=5x+11$
$\displaystyle \Rightarrow x^2-15x+14=0$
$\displaystyle \Rightarrow x^2-14x-x+14=0$
$\displaystyle \Rightarrow x(x-14)-1(x-14)=0$
$\displaystyle \Rightarrow (x-14)(x-1)=0$
$\displaystyle \Rightarrow x=14\text{ or }x=1$
$\displaystyle \text{Since }x=1\text{ would make }(x-5)\text{ negative, }x=1\text{ is rejected.}$
$\displaystyle \therefore \text{Zeba's present age is }14\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{At present Asha's age (in years) is }2\text{ more than the square of her daughter}$
$\displaystyle \text{Nisha's age. When Nisha grows to her mother's present age, Asha's age would be one year}$
$\displaystyle \text{less than }10\text{ times the present age of Nisha. Find the present ages of Asha and Nisha.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let Nisha's present age be }x\text{ years.}$
$\displaystyle \therefore \text{Asha's present age}=(x^2+2)\text{ years.}$
$\displaystyle \text{Nisha will reach Asha's present age after }(x^2+2-x)\text{ years.}$
$\displaystyle \text{Asha's age at that time}=(x^2+2)+(x^2+2-x)$
$\displaystyle =2x^2-x+4\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle 2x^2-x+4=10x-1$
$\displaystyle \Rightarrow 2x^2-11x+5=0$
$\displaystyle \Rightarrow 2x^2-10x-x+5=0$
$\displaystyle \Rightarrow 2x(x-5)-1(x-5)=0$
$\displaystyle \Rightarrow (x-5)(2x-1)=0$
$\displaystyle \Rightarrow x=5\text{ or }x=\frac{1}{2}$
$\displaystyle \text{If }x=\frac{1}{2},\text{ Asha's present age would be }\frac{1}{4}+2=\frac{9}{4}\text{ years, which is impossible.}$
$\displaystyle \therefore x=5.$
$\displaystyle \therefore \text{Nisha's present age}=5\text{ years.}$
$\displaystyle \therefore \text{Asha's present age}=5^2+2=27\text{ years.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{The age of a man is twice the square of the age of his son. Eight years hence, the age}$
$\displaystyle \text{of the man will be }4\text{ years more than three times the age of his son. Find their present ages.}$
$\displaystyle \hfill\text{[CBSE 2024]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the present age of the son be }x\text{ years.}$
$\displaystyle \therefore \text{Present age of the man}=2x^2\text{ years.}$
$\displaystyle \text{Eight years hence, son's age}=(x+8)\text{ years.}$
$\displaystyle \text{Eight years hence, man's age}=(2x^2+8)\text{ years.}$
$\displaystyle \text{According to the question,}$
$\displaystyle 2x^2+8=3(x+8)+4$
$\displaystyle \Rightarrow 2x^2+8=3x+24+4$
$\displaystyle \Rightarrow 2x^2-3x-20=0$
$\displaystyle \Rightarrow 2x^2-8x+5x-20=0$
$\displaystyle \Rightarrow 2x(x-4)+5(x-4)=0$
$\displaystyle \Rightarrow (x-4)(2x+5)=0$
$\displaystyle \Rightarrow x=4\text{ or }x=-\frac{5}{2}$
$\displaystyle \text{Since age cannot be negative, }x=-\frac{5}{2}\text{ is rejected.}$
$\displaystyle \therefore \text{Present age of the son}=4\text{ years.}$
$\displaystyle \therefore \text{Present age of the man}=2(4)^2=32\text{ years.}$
$\displaystyle \\$