Given:

(i) Total marbles with John and Jivanti = $45$. After losing $5$ marbles each, the product of the remaining marbles is $124$.

(ii) Let $x$ be the number of toys produced. The cost of production per toy = $55 - x$. Total cost of production = $Rs. 750$.

To Find:

(i) The initial number of marbles John and Jivanti had.

(ii) The number of toys produced on that day.

Part (i): Solving for Marbles

Step 1: Define variables.

Let the number of marbles John had be $x$.

Since the total number of marbles is $45$, the number of marbles Jivanti had is $(45 - x)$.

Step 2: Formulate the equation based on the condition.

After losing $5$ marbles each:

John's remaining marbles = $x - 5$

Jivanti's remaining marbles = $(45 - x) - 5 = 40 - x$

The product of these is $124$:

$(x - 5)(40 - x) = 124$

Step 3: Simplify the quadratic equation.

$40x - x^2 - 200 + 5x = 124$

$-x^2 + 45x - 200 = 124$

$x^2 - 45x + 324 = 0$ [Rearranging terms to standard form $ax^2 + bx + c = 0$]

Step 4: Solve by factorization.

We need two numbers that multiply to $324$ and add to $-45$. These are $-36$ and $-9$.

$x^2 - 36x - 9x + 324 = 0$

$x(x - 36) - 9(x - 36) = 0$

$(x - 36)(x - 9) = 0$

Therefore, $x = 36$ or $x = 9$.

If John had $36$ marbles, Jivanti had $9$. If John had $9$ marbles, Jivanti had $36$.

Final Answer (i): John and Jivanti started with 36 and 9 marbles respectively.

Part (ii): Solving for Toys

Step 1: Define variables.

Let the number of toys produced be $x$.

Cost of production per toy = $(55 - x)$

Step 2: Formulate the equation.

Total cost = (Number of toys) $\times$ (Cost per toy)

$x(55 - x) = 750$

Step 3: Simplify the quadratic equation.

$55x - x^2 = 750$

$x^2 - 55x + 750 = 0$

Step 4: Solve by factorization.

We need two numbers that multiply to $750$ and add to $-55$. These are $-25$ and $-30$.

$x^2 - 30x - 25x + 750 = 0$

$x(x - 30) - 25(x - 30) = 0$

$(x - 30)(x - 25) = 0$

Therefore, $x = 30$ or $x = 25$.

Final Answer (ii): The number of toys produced on that day was either 25 or 30.

Given: A quadratic equation $kx(x - 2) + 6 = 0$, where $k \neq 0$.

To Find: The value(s) of $k$ such that the quadratic equation has two equal real roots.

Step 1: Simplifying the equation into standard form

The standard form of a quadratic equation is $ax^2 + bx + c = 0$. We expand the given equation:

$kx(x - 2) + 6 = 0$

$kx^2 - 2kx + 6 = 0$

[Distributive property of multiplication over subtraction]

Step 2: Identifying coefficients

Comparing $kx^2 - 2kx + 6 = 0$ with $ax^2 + bx + c = 0$, we identify:

$a = k$

$b = -2k$

$c = 6$

Step 3: Applying the condition for equal roots

For a quadratic equation to have two equal real roots, its discriminant ($D$) must be equal to zero.

Formula: $D = b^2 - 4ac = 0$

Substituting the identified coefficients into the discriminant formula:

$(-2k)^2 - 4(k)(6) = 0$

Step 4: Solving for $k$

$4k^2 - 24k = 0$

[Simplifying the powers and products]

Factor out the common term $4k$:

$4k(k - 6) = 0$

[Using the zero-product property: if $p \cdot q = 0$, then $p=0$ or $q=0$]

Case 1: $4k = 0 \implies k = 0$

Case 2: $k - 6 = 0 \implies k = 6$

Step 5: Verification of validity

If $k = 0$, the equation becomes $0x^2 - 2(0)x + 6 = 0$, which simplifies to $6 = 0$. This is a contradiction and not a quadratic equation. Therefore, $k$ cannot be $0$.

If $k = 6$, the equation becomes $6x^2 - 12x + 6 = 0$. Dividing by 6, we get $x^2 - 2x + 1 = 0$, which is $(x - 1)^2 = 0$, yielding equal roots $x = 1, 1$.

Final Answer: The value of $k$ is 6.

Given: The algebraic equation $(x - 2)(x + 1) = (x - 1)(x + 3)$.

To Find: Determine whether the given equation is a quadratic equation.

Definition: A quadratic equation in the variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$.

Step 1: Expanding the Left-Hand Side (LHS)

The LHS is $(x - 2)(x + 1)$. We apply the distributive property of multiplication over addition: $(a + b)(c + d) = ac + ad + bc + bd$.

$(x - 2)(x + 1) = x(x) + x(1) - 2(x) - 2(1)$

$= x^2 + x - 2x - 2$

$= x^2 - x - 2$

Step 2: Expanding the Right-Hand Side (RHS)

The RHS is $(x - 1)(x + 3)$. Similarly, applying the distributive property:

$(x - 1)(x + 3) = x(x) + x(3) - 1(x) - 1(3)$

$= x^2 + 3x - x - 3$

$= x^2 + 2x - 3$

Step 3: Equating LHS and RHS and Simplifying

Now, set the expanded LHS equal to the expanded RHS:

$x^2 - x - 2 = x^2 + 2x - 3$

To bring all terms to one side, subtract $(x^2 + 2x - 3)$ from both sides:

$x^2 - x^2 - x - 2x - 2 + 3 = 0$

Combine like terms:

$(1 - 1)x^2 + (-1 - 2)x + (-2 + 3) = 0$

$0x^2 - 3x + 1 = 0$

$-3x + 1 = 0$

Step 4: Conclusion

The resulting equation is $-3x + 1 = 0$. This is a linear equation, not a quadratic equation, because the coefficient of $x^2$ is $0$ (i.e., $a = 0$). Since the definition of a quadratic equation requires $a \neq 0$, this equation does not satisfy the condition.

Final Answer: The given equation $(x - 2)(x + 1) = (x - 1)(x + 3)$ is not a quadratic equation.

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.