Question

Question- Write the quadratic equation for the following:“The age of Ruchi is twice the age of Monika. The

product of their ages will be 160 after four years. Find their present ages.”​

Answer 1

Let the age of Monika be x years.

A/Q , Ruchi is twice the age of Monika .

$\therefore$ Ruchi's age = 2x .

After 4 years,

• Monika's age = x +4
• Ruchi's age = 2x + 4

A/Q , product of their ages after 4 years is 160 .

$\therefore$ (x+4) (2x + 4) = 160

→ x ( 2x + 4) + 4(2x + 4) = 160

→ 2x² + 4x + 8x + 16 = 160

→ 2x² + 12x + 16 = 160

Dividing the equation by 2 :

→ x² + 6x + 8 = 80

Hence the required quadratic equation is :

$\boxed{\tt x^2 + 6x + 8 = 80}$

Solving the equation :

x² + 6x + 8 = 80

→ x² + 6x +8 - 80 = 0

→ x² + 6x - 72 = 0

→ x² + ( 12 - 6) x -72 = 0

→ x² + 12x - 6 x - 72 = 0

→ x( x +12) - 6 (x + 12) = 0

→ (x-6) (x +12) = 0

x -6 = 0

→ x = 6

x +12 = 0

→ x = -12

Since age can't be negetive , ignoring the negetive value and taking x as 6 .

• Monika's present age = 6 years
• Ruchi's present age = 2× 6 = 12 years

Answer 2

ANSWER:

According to Problem:

Let the age of Monika be x years.

and, Ruchi's age will be 2x years.

Now,

Monika's age after 4 years = x + 4

And Ruchi's age after 4 years = 2x + 4

ATQ,

Product of their ages is 160 after 4 years.

= $(x + 4) (2x + 4) = 160$

= $x(2x + 4) + 4(2x + 4) = 160$

= $2 {x}^{2} + 4x+ 8x + 16 = 160$

= $2 {x}^{2} + 12x + 16 = 160$

Divide this quadratic equation by 2

${x}^{2} + 6x + 8 = 80$

By solving this equation:

= ${x}^{2} + 6x + 8 = 80$

= ${x}^{2} + 6x + 8 - 80 = 0$

= ${x}^{2} + 6x - 72 = 0$

= ${x}^{2} + (12 - 6)x - 72 = 0$

= ${x}^{2} + 12x - 6x - 72 = 0$

= $x(x + 12) - 6(x + 12) = 0$

= $(x - 6)(x + 12) = 0$

= $(x - 6) = 0 \: or \: (x + 12) = 0$

= $x = 6 \: or \: x = - 12$

Note: As age cannot be negative, ignore the negative value and take x as 6.

Hence,

Monika's present age = x = 6 years.

Ruchi's present age = 2x = 2 × 6 = 12 years.

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Answered By : MoonGurl01