NCERT · Class 10 · Mathematics · Quadratic EquationsA motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. Also, explain the theoretical concept of the discriminant of a quadratic equation and how it helps in determining the nature of roots.

Part 1: Numerical Problem

Given:

• Speed of the motor boat in still water = $18\text{ km/h}$
• Let the speed of the stream be $x\text{ km/h}$
• Therefore, the speed of the boat upstream = $(18 - x)\text{ km/h}$
• The speed of the boat downstream = $(18 + x)\text{ km/h}$
• Distance = $24\text{ km}$

Formula: $$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$$ \nAccording to the question, the time taken to go upstream is $1\text{ hour}$ more than the time taken to go downstream: $$\frac{24}{18 - x} - \frac{24}{18 + x} = 1$$

Step-by-step Solution:

1. Take $24$ as a common factor: $$24 \left( \frac{1}{18 - x} - \frac{1}{18 + x} \right) = 1$$

2. Find the common denominator for the terms inside the bracket: $$24 \left[ \frac{(18 + x) - (18 - x)}{(18 - x)(18 + x)} \right] = 1$$

3. Simplify the numerator and denominator (using $(a-b)(a+b) = a^2 - b^2$): $$24 \left[ \frac{18 + x - 18 + x}{18^2 - x^2} \right] = 1$$ $$24 \left[ \frac{2x}{324 - x^2} \right] = 1$$

4. Multiply the numerator: $$\frac{48x}{324 - x^2} = 1$$

5. Cross-multiply to form a standard quadratic equation: $$48x = 324 - x^2$$ $$x^2 + 48x - 324 = 0$$

6. Solve the quadratic equation by splitting the middle term. We need two numbers whose product is $-324$ and sum is $48$. These numbers are $54$ and $-6$: $$x^2 + 54x - 6x - 324 = 0$$ $$x(x + 54) - 6(x + 54) = 0$$ $$(x - 6)(x + 54) = 0$$

7. This gives two possible values for $x$: $$x = 6 \quad \text{or} \quad x = -54$$ \nSince the speed of the stream cannot be negative, we discard $x = -54$.

Conclusion:\nThe speed of the stream is $6\text{ km/h}$.

Part 2: Theoretical Concept

The Discriminant of a Quadratic Equation \nA quadratic equation in its standard form is written as: $$ax^2 + bx + c = 0 \quad (where \ a \neq 0)$$ \nThe expression $b^2 - 4ac$ is known as the discriminant of the quadratic equation, usually denoted by the symbol $D$ or $\Delta$: $$D = b^2 - 4ac$$

Role in Determining the Nature of Roots:\nThe discriminant plays a crucial role in predicting the nature of the roots of a quadratic equation without actually calculating the roots themselves, by using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

1. Case 1: If $D > 0$ (Positive Discriminant)

   • The quadratic equation has two distinct real roots.
   • If $b^2 - 4ac$ is a perfect square, the roots are rational numbers.
   • If it is not a perfect square, the roots are irrational and occur in conjugate pairs.

2. Case 2: If $D = 0$ (Zero Discriminant)

   • The quadratic equation has two equal real roots (or one repeated real root).
   • The roots are given by $x = \frac{-b}{2a}$.

3. Case 3: If $D < 0$ (Negative Discriminant)

   • The quadratic equation has no real roots because the square root of a negative number is not a real number in the real number system.
   • In higher mathematics, these are studied as complex or imaginary roots.