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Q4:
Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is $20$ years. Four years ago, the product of their ages in years was $48$.
Solution :
Given:
1. The sum of the ages of two friends is $20$ years.
2. Four years ago, the product of their ages was $48$.
To Find:
Determine if the situation is possible, and if so, find the present ages of the two friends.
Step 1: Defining Variables
Let the present age of the first friend be $x$ years.
Since the sum of their ages is $20$ years, the present age of the second friend is $(20 - x)$ years.
Step 2: Formulating the Equation based on the condition "Four years ago"
Age of the first friend four years ago = $(x - 4)$ years.
Age of the second friend four years ago = $(20 - x - 4) = (16 - x)$ years.
According to the problem, the product of these ages is $48$:
$(x - 4)(16 - x) = 48$
Step 3: Expanding and Simplifying the Equation
Expanding the left side using the distributive property:
$x(16) - x(x) - 4(16) + 4(x) = 48$
$16x - x^2 - 64 + 4x = 48$
$-x^2 + 20x - 64 = 48$
Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:
$-x^2 + 20x - 64 - 48 = 0$
$-x^2 + 20x - 112 = 0$
Multiplying by $-1$ to make the leading coefficient positive:
$x^2 - 20x + 112 = 0$
Step 4: Checking for the possibility of the situation
To determine if the situation is possible, we calculate the discriminant ($D$) of the quadratic equation $ax^2 + bx + c = 0$, where $a = 1$, $b = -20$, and $c = 112$.
The formula for the discriminant is $D = b^2 - 4ac$.
$D = (-20)^2 - 4(1)(112)$
$D = 400 - 448$
$D = -48$
Step 5: Conclusion based on the Discriminant
[Since the discriminant $D < 0$, the quadratic equation has no real roots.]
Because the roots are not real, it is impossible to find real values for the ages of the friends that satisfy the given conditions.
Final Answer: The situation is not possible.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.3
- Q1(i): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2x^2 – 3x + 5 = 0$
- Q1(ii): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (ii) $3x^2 – 4\sqrt{3}x + 4 = 0$
- Q1(iii): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (iii) $2x^2 – 6x + 3 = 0$
- Q2(i): Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (i) $2x^2 + kx + 3 = 0$
- Q2(ii): Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (ii) $kx (x – 2) + 6 = 0$
- Q3: Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800$ $m^2$? If so, find its length and breadth.
- Q5: Is it possible to design a rectangular park of perimeter $80$ m and area $400$ $m^2$? If so, find its length and breadth.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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