Question

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages was 48 years. Frame the equation for the above situation and predict the nature of its roots.

• A. $x^2+20x+112=0$ The roots are real and negative
• B. $x^2-20x-112=0$ The roots are real and positive
• C. $x^2+20x-112=0$ The roots are real and equal
• D. $x^2-20x+112=0$ No real roots

Answer 1

The correct option is D

$x^2-20x+112=0$

No real roots

Let the age of one friend = x years
Age of the other friend = 20 - x years

Four years ago,
Age of one friend = (x - 4) years
Age of the other friend = (20 - x - 4) years

According to the condition, (x - 4)(16 - x) = 48
$\Rightarrow 16x-64-x^2+4x=48$
$\Rightarrow -x^2+20x-64-48=0$
$\Rightarrow -x^2+20x-112=0$
$\Rightarrow x^2-20x+112=0$

Comparing the given equation to the standard form, we get, a = 1, b = -20, c = 112
$\therefore b^2-4ac=(-20{)}^2-4(1\left)\right(112)=400-448=-48<0$

Since $b^2-4ac<0$, the quadratic equation has no real roots.
$\therefore$ The given situation is not possible