Quadratic Equations

We are given the equation: where , and and are integers.
Rearrange the equation to express : Now, we will substitute different integer values for such that . 1. If : Thus, when , , which satisfies .

The correct option is (D):

For a quadratic equation , the nature of its roots is determined by the discriminant . The roots are:
Real and rational if and is a perfect square.
Real and irrational if and is not a perfect square.
Complex (imaginary) if .

Since the roots are given to be irrational, this means: and is not a perfect square.

Hence, the correct answer is that the discriminant is not a perfect square.

By the , the remainder when is divided by is .
Here, we divide by , so we compute :

The correct answer is option (A):

Correct answer: 4

Explanation:
The given equation is: Simplify the left-hand side: So the equation becomes: Move all terms to one side: Simplify: The resulting polynomial is: This has degree 2, but the original equation before simplification had terms up to on both sides. However, when solving an equation, the degree is based on the highest power of the variable in the simplified form of the equation. On the right-hand side: → highest power is
On the left-hand side: → again highest power is Therefore, degree of the equation is 4.

Correct answer: product of two consecutive positive integers is 306

Explanation:
Let the two consecutive positive integers be and .
Their product is: Given: This matches the given equation:

Hence, the equation represents the product of two consecutive positive integers is 306.

Correct option:
is a quadratic equation.

Explanation:
Let's simplify each option one by one:

1. → This expression is incomplete or incorrect (possibly a typo). So, it is not a valid equation.

2.
LHS:
RHS:
So, the equation becomes
Subtracting from both sides:

This is a linear equation.

3.
Expand both terms:


Still a linear equation (not quadratic).

4.
LHS:
RHS:
So, the equation becomes
This is an identity, not a quadratic equation.

Hence, none of the options represents a proper quadratic equation in the standard form .
However, if the first option was meant to be: , then:

This is a quadratic equation.

Final Answer: If corrected, option 1: is the quadratic equation.

Correct answer: infinitely many solutions

Explanation:
The equation x - 4y = 5 is a linear equation in two variables.

A linear equation in two variables represents a straight line on the coordinate plane and has infinitely many solutions, since there are infinitely many points (x, y) that satisfy the equation.

Therefore, the given equation has infinitely many solutions.

Correct answer: 1

Explanation:
The given polynomial is: x + 2x² + 1

Rearranging in standard form: 2x² + x + 1

For a quadratic polynomial ax² + bx + c, the product of the zeroes is given by: Product = c / a

Here,
a = 2, b = 1, c = 1

So, Product = 1 / 2 = 0.5

Hence, the correct answer is: 1/2

The given points are and , and the point divides the line segment joining these points in the ratio 1:2 internally.
The formula to find the coordinates of a point dividing the line segment in the ratio is:
Substitute the values: The coordinates of the dividing point are approximately . This point lies in the 4th quadrant.

Therefore, the correct answer is (D): 4^th quadrant

The general form of the -th term of a geometric progression (GP) is given by: where is the first term and is the common ratio. From the question: - The first term so, . - The fourth term , so we have the equation: Substituting : Taking the cube root of both sides:

The correct option is (C):

The given arithmetic progression (AP) is:
.
This is an arithmetic progression with the first term and common difference .
The formula for the -th term of an AP is: \\ For the 15th term, substitute , , and : Thus, the 15th term of the AP is .

The correct option is (B):

Let the age of the father be and the age of the son be . We are given the following two conditions: 1. The sum of their ages is 65 years: 2. Twice the difference of their ages is 50 years: Simplify the second equation: Now, we have the system of equations: 1. 2. To solve this system, add the two equations: Thus, the age of the father is 45 years.

The correct option is (A):

We are given the quadratic equation:

One root is given as . Let the other root be .

By Vieta's formulas, the sum of the roots is given by:

Solving for :

Thus, the other root is: .

We are given the quadratic equation:

Let the roots be and .

From Vieta's formulas:

• Sum of the roots:
• Product of the roots:

Substituting values:

We are given that the sum and product of the roots are equal:

Solving for :

Thus, the value of is: .

Step 1: Expand Each Given Equation

Option 1:

Expanding the left-hand side:

So the equation becomes:

Subtracting from both sides:

Since this contains initially, it is not a quadratic equation.

Option 2:

Expanding both sides:

Rearranging:

This is a quadratic equation.

Option 3:

Expanding:

Rearranging:

This is a quadratic equation.

Option 4:

Expanding both sides:

Cancel from both sides:

Rearrange:

This is a linear equation, not a quadratic equation.

Final Answer:

1. Quadratic Formula:

The quadratic formula is .

2. Simplify using given information:

We know :

Therefore, .

3. Find the other root:

Substituting into the original equation:

Since and , we have , so .

The equation becomes

Factoring:

This is a perfect square, meaning the only root is . It's a repeated root.

1. Identify coefficients:

In the equation , we have , , and .

2. Sum of roots:

Sum =

3. Product of roots:

Product =

4. Calculate the ratio:

Ratio = (Sum) : (Product) =