Quadratic Equations

To solve this problem, we need to find the roots of the quadratic equation:

1. Given Equation:

2. Factoring the Quadratic:
We look for two numbers whose product is -6 and sum is +1 (coefficient of x).
The numbers 3 and -2 satisfy this:

3. Writing the Factored Form:

4. Solving for Roots:
Setting each factor to zero gives:

Final Answer:
Option (A) 2, -3 is correct.

To solve this problem, we are given that and are the roots of the quadratic equation , and we are asked to find the value of .

1. Using Root Relationships:
For the quadratic equation , the sum and product of the roots are given by:

2. Using the Identity:
We use the identity:

3. Substituting the Values:

4. Evaluating the Options:

• (1) – Incorrect
• (2) – Incorrect
• (3) – Correct
• (4) – Incorrect

Final Answer:
The correct answer is (C) .

To solve the problem, we need to determine the value of such that the quadratic equation has a root .

1. Substituting the Root into the Equation:
If is a root of the equation, then substituting into the equation should satisfy it. The given equation is:

Substitute :

This simplifies to:

2. Solving for :
Combine like terms:

Solve for :

3. Verifying the Solution:
To ensure correctness, substitute back into the original equation and check if is indeed a root:

Factorize the quadratic equation:

This confirms that is a root.

Final Answer:
The value of is .

To solve the problem, we need to determine the value of given that the roots of the quadratic equation are and .

1. Using Vieta's Formulas:
For a quadratic equation with roots and , Vieta's formulas tell us:

2. Expressing and :
We know from trigonometric identities that: Since , we can rewrite this as: Substituting and , we get:

3. Solving for :
From the equation above, we directly see that:

Final Answer:
The value of is .

To solve the problem, we need to determine the nature (sign) of the zeroes of the quadratic polynomial .

1. Recall the standard form of a quadratic equation:
A quadratic polynomial can be written as:


For our equation, , , and .

2. Find the discriminant:
The discriminant of a quadratic equation is given by:


Substitute the values:


Since , the roots are real and distinct.

3. Find the roots using factorization:
We need to find two numbers that multiply to 119 and add to 24.
The numbers are 17 and 7, so we can factor the equation as:





Therefore, the roots are and .

4. Determine the signs of the roots:
Both roots are negative.

Final Answer:
The correct answer is option (C): both negative.

To solve this problem, we need to determine the cardinal number of the given set .

1. Understanding Cardinal Number:
The cardinal number of a set refers to the number of distinct elements present in that set.

2. Counting the Elements:
The set contains the elements: -2, -1, 0, 1, 2
So, total elements = 5

3. Evaluating the Options:
(1) 5 – Correct
(2) 4 – Incorrect
(3) -2 – Incorrect (cardinal number is never negative)
(4) 2 – Incorrect

Final Answer:
The correct option is (A) 5.

We are given a linear equation in one variable:

Step 1: Subtract from both sides of the equation

Step 2: Divide both sides by to isolate

The correct option is (D):

Step 1: Recall the standard form of a quadratic equation

The general form is:

Step 2: Identify coefficients from the given equation

Given:
Here, , ,

Step 3: Use the formula for sum of roots

Sum of roots of a quadratic equation is given by:

Step 4: Substitute the values of and

Correct Option: (A): 2

Step 1: For a quadratic equation of the form , the condition for equal roots is that the discriminant is zero.

Step 2: The discriminant is given by:

Step 3: In the given equation, compare with standard form:


Step 4: Apply the condition for equal roots:


Step 5: Hence, the value of is

The correct option is (C):

We are given the quadratic equation:

Step 1: For the quadratic equation to have real roots, its discriminant must be greater than or equal to zero.
The discriminant is given by:


Step 2: Compare with the general form:

So, the discriminant becomes:


Step 3: For real roots, :

⇒
⇒

Therefore, the correct condition is (C) :

Step 1: Definition of a quadratic equation

A quadratic equation is a polynomial equation of degree 2. The general form is: where and most importantly, .

Step 2: Why must ?

If , the equation becomes: which is a linear equation (not quadratic). So, to ensure the term exists, must not be zero.

Step 3: Role of and

The values of and can be zero, non-zero, or any real number. They don’t affect the degree of the equation. Only guarantees it is quadratic.

The correct option is (A):

Step 1: Understand the condition for no solution.

Two linear equations of the form: have no solution if: This is the condition for inconsistency — the lines are parallel but not coincident.

Step 2: Match coefficients.

Given equations:

• Equation 1:
• Equation 2:

Here,

Now, apply the condition:

Step 3: Solve for

From , cross-multiply:

Step 4: Check that

And:

The correct option is (A):