A 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 t

Question

A 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.

Accepted Answer

Part 1: Numerical Problem Given: Speed of the motor boat in still water = $18	ext{ km/h}$ Let the speed of the stream be $x	ext{ km/h}$ Therefore, the speed of the boat upstream = $(18 - x)	ext{ km/h}$ The speed of the boat downstream = $(18 + x)	ext{ km/h}$ Distance = $24	ext{ km}$ Formula: $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}}$$ 
According to the question, the time taken to go upstream is $1	ext{ hour}$ more than the time taken to go downstream: $$\frac{24}{18 - x} - \frac{24}{18 + x} = 1$$ Step-by-step Solution: Take $24$ as a common factor: $$24 \left( \frac{1}{18 - x} - \frac{1}{18 + x} ight) = 1$$ Find the common denominator for the terms inside the bracket: $$24 \left \frac{(18 + x) - (18 - x)}{(18 - x)(18 + x)} ight = 1$$ Simplify the numerator and denominator (using $(a-b)(a+b) = a^2 - b^2$): $$24 \left \frac{18 + x - 18 + x}{18^2 - x^2} ight = 1$$ $$24 \left \frac{2x}{324 - x^2} ight = 1$$ Multiply the numerator: $$\frac{48x}{324 - x^2} = 1$$ Cross-multiply to form a standard quadratic equation: $$48x = 324 - x^2$$ $$x^2 + 48x - 324 = 0$$ 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 $-6$: $$x^2 + 54x - 6x - 324 = 0$$ $$x(x + 54) - 6(x + 54) = 0$$ $$(x - 6)(x + 54) = 0$$ This gives two possible values for $x$: $$x = 6 \quad 	ext{or} \quad x = -54$$ 
Since the speed of the stream cannot be negative, we discard $x = -54$. Conclusion:
The speed of the stream is $6	ext{ km/h}$. --- Part 2: Theoretical Concept The Discriminant of a Quadratic Equation 
A quadratic equation in its standard form is written as: $$ax^2 + bx + c = 0 \quad (where \ a 
eq 0)$$ 
The 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:
The 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: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 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. 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 $x = \frac{-b}{2a}$. Case 3: If $D

Quadratic Equations

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