Respuesta :
If the current son's age is x, the current age of the father is [tex]x^2[/tex].
In 20 years they will be, respectively, [tex]x+20[/tex] and [tex]x^2+20[/tex] years old, and we know that the father will be 3 times as old as his son:
[tex]x^2+20=3(x+20) \iff x^2+20=3x+60 \iff x^2-3x-40=0[/tex]
The quadratic equation has solutions
[tex]x_{1,2}=\dfrac{3\pm\sqrt{9+160}}{2}=\dfrac{3\pm 13}{2}[/tex]
So, the two solutions are
[tex]x_1 = \dfrac{16}{2}=8,\quad x_2 = \dfrac{-10}{2}=-5[/tex]
We clearly can't accept negative values as solution for a problem involving the age of a person, so the only feasible solution is x=8, and we deduce that the father's age is 8 squared, i.e. 64.
Present age of the father is 64 years and age of the son is 8 years.
Let the present age of father = y years
And the present age of son = x years
Now first statement states, "father's age is the square of his son's age".
Equation by this statement will be,
y = x² ---------(1)
Second statement states, "After 20 years, father will be 3 times as old as his son"
So the equation will be,
(y + 20) = 3(x + 20)
y + 20 = 3x + 60
y = 3x + 60 - 20
y = 3x + 40 -------------(2)
By substituting the value of y in equation (2) from equation (1),
x² = 3x + 40
x² - 3x - 40 = 0
x² - 8x + 5x - 40 = 0
x(x - 8) + 5(x - 8) = 0
(x + 5)(x - 8) = 0
x = -5, 8
But age can't be negative.
Therefore, x = 8 years
By substituting the value of x in equation (1),
y = 8²
y = 64 years
Therefore, present age of the father is 64 years and age of the son is 8 years.
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