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:

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.

Given:

1. The shape of the mango grove is rectangular.

2. The length ($l$) of the grove is twice its breadth ($b$), i.e., $l = 2b$.

3. The area ($A$) of the rectangular grove is $800$ $m^2$.

To Find:

1. Determine if such a rectangular grove can be designed.

2. If possible, find the length ($l$) and breadth ($b$).

Visual Representation:

Step 1: Formulating the Algebraic Equation

Let the breadth of the rectangular mango grove be $b$ meters.

According to the problem, the length $l$ is twice the breadth, so $l = 2b$ meters.

The formula for the area of a rectangle is: $Area = length \times breadth$.

Substituting the given values into the formula:

$800 = (2b) \times (b)$

$800 = 2b^2$

Step 2: Simplifying the Quadratic Equation

Divide both sides of the equation by $2$:

$\frac{800}{2} = \frac{2b^2}{2}$

$400 = b^2$

Rearranging into the standard quadratic form $ax^2 + bx + c = 0$:

$b^2 - 400 = 0$

Step 3: Determining Possibility using the Discriminant

To check if real roots exist, we calculate the discriminant ($D$) where $D = B^2 - 4AC$.

In the equation $b^2 + 0b - 400 = 0$, we have $A = 1$, $B = 0$, and $C = -400$.

$D = (0)^2 - 4(1)(-400)$

$D = 0 + 1600$

$D = 1600$

[Since $D > 0$, the quadratic equation has two distinct real roots, meaning it is possible to design the grove.]

Step 4: Solving for Breadth and Length

From $b^2 = 400$, we take the square root of both sides:

$b = \pm \sqrt{400}$

$b = \pm 20$

[Since breadth cannot be negative, we discard $b = -20$.]

Therefore, $b = 20$ meters.

Now, calculate the length $l$ using $l = 2b$:

$l = 2 \times 20 = 40$ meters.

Final Answer:

Yes, it is possible to design the mango grove. The breadth of the grove is $20$ m and the length is $40$ m.

Given: A quadratic equation $2x^2 - 6x + 3 = 0$.

To Find: The nature of the roots of the given quadratic equation and, if real roots exist, find their values.

Step 1: Identify the coefficients of the quadratic equation.

A standard quadratic equation is represented as $ax^2 + bx + c = 0$. Comparing the given equation $2x^2 - 6x + 3 = 0$ with the standard form:

• $a = 2$
• $b = -6$
• $c = 3$

Step 2: Determine the nature of the roots using the Discriminant ($D$).

The discriminant of a quadratic equation is given by the formula $D = b^2 - 4ac$.

Substituting the values of $a$, $b$, and $c$:

$D = (-6)^2 - 4(2)(3)$

$D = 36 - 24$

$D = 12$

[Since $D > 0$, the quadratic equation has two distinct real roots.]

Step 3: Apply the Quadratic Formula to find the roots.

The quadratic formula is given by $x = \frac{-b \pm \sqrt{D}}{2a}$.

Substituting the known values into the formula:

$x = \frac{-(-6) \pm \sqrt{12}}{2(2)}$

$x = \frac{6 \pm \sqrt{4 \times 3}}{4}$

$x = \frac{6 \pm 2\sqrt{3}}{4}$

Step 4: Simplify the expression.

Factor out the common term $2$ from the numerator:

$x = \frac{2(3 \pm \sqrt{3})}{4}$

$x = \frac{3 \pm \sqrt{3}}{2}$

Therefore, the two roots are:

$x_1 = \frac{3 + \sqrt{3}}{2}$

$x_2 = \frac{3 - \sqrt{3}}{2}$

Final Answer: The roots are real and distinct. The roots of the equation are $\frac{3 + \sqrt{3}}{2}$ and $\frac{3 - \sqrt{3}}{2}$.

Given: A quadratic equation $3x^2 - 4\sqrt{3}x + 4 = 0$.

To Find: The nature of the roots and the roots themselves if they exist.

Step 1: Identify the coefficients of the quadratic equation.
The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Comparing the given equation $3x^2 - 4\sqrt{3}x + 4 = 0$ with the standard form, we identify:
$a = 3$
$b = -4\sqrt{3}$
$c = 4$

Step 2: Determine the nature of the roots using the Discriminant ($D$).
The discriminant is given by the formula $D = b^2 - 4ac$.
Substituting the values identified in Step 1:
$D = (-4\sqrt{3})^2 - 4(3)(4)$
$D = (16 \times 3) - 48$
$D = 48 - 48$
$D = 0$
[Since $D = 0$, the quadratic equation has two equal real roots.]

Step 3: Calculate the roots using the Quadratic Formula.
The quadratic formula is given by $x = \frac{-b \pm \sqrt{D}}{2a}$.
Since $D = 0$, the formula simplifies to $x = \frac{-b}{2a}$.
Substituting the values:
$x = \frac{-(-4\sqrt{3})}{2(3)}$
$x = \frac{4\sqrt{3}}{6}$

Step 4: Simplify the expression.
$x = \frac{4\sqrt{3}}{6}$
Dividing both the numerator and the denominator by their greatest common divisor, which is $2$:
$x = \frac{2\sqrt{3}}{3}$
Since the roots are equal, both roots are $\frac{2\sqrt{3}}{3}$.
[Note: $\frac{2\sqrt{3}}{3}$ can also be written as $\frac{2}{\sqrt{3}}$ by rationalizing the denominator.]

Final Answer: The roots are real and equal, and the roots are $\frac{2\sqrt{3}}{3}, \frac{2\sqrt{3}}{3}$.

Given: A quadratic equation $2x^2 - 3x + 5 = 0$.

To Find: The nature of the roots of the given quadratic equation and, if real roots exist, to determine their values.

Step 1: Identify the coefficients of the quadratic equation.
A standard quadratic equation is represented as $ax^2 + bx + c = 0$.
Comparing the given equation $2x^2 - 3x + 5 = 0$ with the standard form:
$a = 2$
$b = -3$
$c = 5$

Step 2: Determine the nature of the roots using the Discriminant ($D$).
The discriminant of a quadratic equation is given by the formula:
$D = b^2 - 4ac$
Substituting the values of $a$, $b$, and $c$ into the formula:
$D = (-3)^2 - 4(2)(5)$
$D = 9 - 40$
$D = -31$

Step 3: Analyze the nature of the roots based on the value of $D$.
According to the theory of quadratic equations:
1. If $D > 0$, the equation has two distinct real roots.
2. If $D = 0$, the equation has two equal real roots.
3. If $D < 0$, the equation has no real roots (the roots are complex/imaginary).

Since $D = -31$, and $-31 < 0$, the discriminant is negative.

Step 4: Conclusion regarding the existence of real roots.
Because the discriminant is less than zero ($D < 0$), the square root of the discriminant would result in an imaginary number. Therefore, the quadratic equation $2x^2 - 3x + 5 = 0$ does not possess any real roots.

Final Answer: The discriminant is $-31$, which is less than $0$. Therefore, the given quadratic equation has no real roots.

Given: A quadratic equation $2x^2 + kx + 3 = 0$.

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

Step 1: Identify the coefficients of the quadratic equation.

A standard quadratic equation is represented as $ax^2 + bx + c = 0$. Comparing the given equation $2x^2 + kx + 3 = 0$ with the standard form, we identify:

$a = 2$

$b = k$

$c = 3$

Step 2: State the condition for equal roots.

For a quadratic equation $ax^2 + bx + c = 0$, the nature of the roots is determined by the discriminant, denoted by $D$, where $D = b^2 - 4ac$.

The condition for a quadratic equation to have two equal real roots is that the discriminant must be equal to zero:

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

Step 3: Substitute the coefficients into the discriminant formula.

Substituting $a = 2$, $b = k$, and $c = 3$ into the equation $b^2 - 4ac = 0$:

$(k)^2 - 4(2)(3) = 0$

Step 4: Solve the resulting equation for $k$.

Perform the multiplication inside the expression:

$k^2 - 24 = 0$

Isolate $k^2$ by adding 24 to both sides of the equation:

$k^2 = 24$

Take the square root of both sides to solve for $k$:

$k = \pm\sqrt{24}$

Simplify the radical expression $\sqrt{24}$:

$\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

Therefore:

$k = \pm 2\sqrt{6}$

Step 5: Conclusion.

The values of $k$ for which the quadratic equation $2x^2 + kx + 3 = 0$ has two equal roots are $2\sqrt{6}$ and $-2\sqrt{6}$.

Final Answer: k = 2\sqrt{6}, -2\sqrt{6}

Given:

Perimeter of the rectangular park ($P$) = $80$ m.

Area of the rectangular park ($A$) = $400$ m$^2$.

To Find:

Determine if such a park can be designed, and if so, find its length ($l$) and breadth ($b$).

Step 1: Formulating the Equations

Let the length of the rectangular park be $l$ meters and the breadth be $b$ meters.

The formula for the perimeter of a rectangle is $P = 2(l + b)$.

Given $P = 80$, we have:

$2(l + b) = 80$

$l + b = 40$

$b = 40 - l$ --- (Equation 1)

The formula for the area of a rectangle is $A = l \times b$.

Given $A = 400$, we have:

$l \times b = 400$ --- (Equation 2)

Step 2: Substituting Equation 1 into Equation 2

Substitute $b = 40 - l$ into the area equation:

$l(40 - l) = 400$

$40l - l^2 = 400$

Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:

$l^2 - 40l + 400 = 0$

Step 3: Checking the Discriminant

To determine if the roots are real, we calculate the discriminant ($D = b^2 - 4ac$).

Here, $a = 1$, $b = -40$, and $c = 400$.

$D = (-40)^2 - 4(1)(400)$

$D = 1600 - 1600$

$D = 0$

[Since $D = 0$, the quadratic equation has two equal real roots, meaning it is possible to design such a park.]

Step 4: Solving for $l$

Using the quadratic formula $l = \frac{-b \pm \sqrt{D}}{2a}$:

$l = \frac{-(-40) \pm \sqrt{0}}{2(1)}$

$l = \frac{40}{2}$

$l = 20$ m

Step 5: Finding the breadth $b$

Substitute $l = 20$ into Equation 1:

$b = 40 - 20$

$b = 20$ m

Conclusion:

Since the length and breadth are equal, the rectangular park is a square with side length $20$ m.

Final Answer: Yes, it is possible to design the park. The length is 20 m and the breadth is 20 m.