Quadratic Equations

Concept: For a quadratic equation , the nature of its roots is determined by the discriminant, .
If , the roots are real and distinct (unequal).
If , the roots are real and equal (repeated).
If , the roots are complex (not real). The question states that the equation has "real and distinct roots". Step 1: Identify coefficients a, b, and c from the given equation The given quadratic equation is . Comparing with :


Step 2: Apply the condition for real and distinct roots For real and distinct roots, the discriminant must be greater than zero: Step 3: Substitute the coefficients into the discriminant inequality Step 4: Solve the inequality for k Divide by 4 (which is a positive number, so the inequality sign does not change): This can also be written as: The condition for the equation to have real and distinct roots is . This matches option (1).

Concept: For a quadratic polynomial , if and are its zeros, then:
Sum of zeros:
Product of zeros: We also use the algebraic identity: . Step 1: Identify coefficients a, b, and c from the given polynomial The given polynomial is . Comparing with :


Step 2: Calculate the sum of the zeros ( ) Step 3: Calculate the product of the zeros ( ) Step 4: Use the identity to find The identity is . Substitute the values of and : To subtract, find a common denominator (which is 4): The value of is . This matches option (4).

Concept: For a quadratic polynomial , let and be its zeros.
Sum of zeros:
Product of zeros:
Discriminant: . If , zeros are real and distinct. If , zeros are real and equal. If , zeros are complex (not real). The signs of the sum and product of zeros can tell us about the signs of the zeros themselves. Step 1: Identify coefficients a, b, and c The given polynomial is . Comparing with :


Step 2: Calculate the sum of the zeros ( ) Since the sum of the zeros is negative ( ). Step 3: Calculate the product of the zeros ( ) Since the product of the zeros is positive ( ). Step 4: Analyze the signs of the zeros based on their sum and product
If the product is positive, it means either both zeros are positive OR both zeros are negative.
If the product is negative, it means one zero is positive and the other is negative. In our case, (positive). So, either both zeros are positive or both are negative. Now consider the sum (negative).
If both zeros were positive, their sum would be positive. This contradicts .
If both zeros are negative, their sum would be negative. This is consistent with . Therefore, both zeros must be negative. Step 5: Check the discriminant (optional, to ensure real zeros) . Since , the zeros are real and distinct. Since , the zeros are not equal. Conclusion: The zeros are both negative. This matches option (2).

Concept: If and are the zeroes of a quadratic polynomial, then the polynomial can be written in the form , where is any non-zero constant. Step 1: Identify the given zeroes Let the zeroes be and . Step 2: Calculate the sum of the zeroes ( ) Step 3: Calculate the product of the zeroes ( ) Step 4: Form a quadratic polynomial using the sum and product of zeroes Using the form , one such polynomial is: Any non-zero constant multiple of this polynomial will also have the same zeroes. So, any polynomial of the form where is a valid answer. Step 5: Check the given options to see which one matches this form or is a multiple
(1) : The constant term is +12, but we need -12. Incorrect.
(2) : Both the coefficient of x and the constant term are incorrect. Incorrect.
(3) : This can be rewritten by factoring out : This is in the form with . Therefore, this polynomial has zeroes -3 and 4. This option is correct.
(4) : This can be rewritten by factoring out 2: The coefficient of x is +1, but we need -1 inside the parenthesis. Incorrect. (If it were , it would be and also correct). Thus, option (3) is a quadratic polynomial whose zeroes are -3 and 4.

Concept: For a quadratic equation , if and are its roots, then:
Sum of roots:
Product of roots: We need to simplify the given expression in terms of and . Step 1: Find the sum and product of the roots for the given equation The equation is . Here, , , . Sum of roots: . Product of roots: . Step 2: Simplify the expression to be evaluated The expression is . To add these fractions, find a common denominator. The common denominator is . This can be written as: Step 3: Substitute the values of and We found: Substitute these into the simplified expression: First, calculate : Now the expression becomes: To divide by a fraction, multiply by its reciprocal: The value of the expression is 8.

Concept: For a quadratic equation , the roots are equal if and only if its discriminant ( ) is equal to zero. The discriminant is given by . Step 1: Identify A, B, and C from the given quadratic equation The given equation is . Comparing with :


Step 2: Set the discriminant to zero Since the equation has equal roots, . Step 3: Expand and simplify the equation Divide the entire equation by 4: Expand : Expand : Now substitute these back into the equation: Cancel out terms: cancels with , and cancels with . We are left with: Multiply by -1 to make the squared terms positive (optional, but helps in recognizing a pattern): Step 4: Factor the resulting expression The expression is a perfect square trinomial. It can be written as . This is of the form , where and . So, . Step 5: Solve for the condition If , then taking the square root of both sides gives: This is the condition for equal roots.

Concept: For a quadratic equation to have real roots, its discriminant ( ) must be greater than or equal to zero ( ). The discriminant is . Step 1: Identify A, B, C for the given equation form The equation is . Here, , , . Step 2: Set up the condition for real roots For real roots, . Substitute the values: Step 3: Test possible values for and Given that and can take values from \{1, 2, 3, 4\}. We need to find pairs that satisfy . Let's iterate through values of :
If : We need . Possible values for from \{1, 2, 3, 4\}:
. ( ) - No
. ( ) - Yes. Pair: (a=1, b=2)
. ( ) - Yes. Pair: (a=1, b=3)
. ( ) - Yes. Pair: (a=1, b=4) (3 pairs for a=1)
If : We need . Possible values for from \{1, 2, 3, 4\}:
. ( ) - No
. ( ) - No
. ( ) - Yes. Pair: (a=2, b=3)
. ( ) - Yes. Pair: (a=2, b=4) (2 pairs for a=2)
If : We need . Possible values for from \{1, 2, 3, 4\}:
. ( ) - No
. ( ) - No
. ( ) - No
. ( ) - Yes. Pair: (a=3, b=4) (1 pair for a=3)
If : We need . Possible values for from \{1, 2, 3, 4\}:
. ( ) - No
. ( ) - No
. ( ) - No
. ( ) - Yes. Pair: (a=4, b=4) (1 pair for a=4) Step 4: Count the total number of valid pairs Total number of equations = (pairs for a=1) + (pairs for a=2) + (pairs for a=3) + (pairs for a=4) Total = . There are 7 such equations. The image shows option (3) 6 circled. Let me recheck the counts. a=1: b=2,3,4 (3 pairs) a=2: b=3,4 (2 pairs) a=3: b=4 (1 pair) a=4: b=4 (1 pair) Total = 3+2+1+1 = 7. My calculation consistently gives 7. If the answer is 6, one of these pairs must be excluded, or one condition was missed. For example, if "real and distinct roots" was implied ( ), then cases where would be excluded. cases: (a=1, b=2) . D=0. (a=4, b=4) . D=0. If D>0 is required, we exclude these 2 pairs, leaving 7-2 = 5 pairs. This does not match 6. The question states "real roots," which includes (equal real roots) and (distinct real roots). So my count of 7 should be correct for "real roots." Given the circled option in the image for many other problems has been correct, if the answer key says 6, there's a subtle point or a common misinterpretation I might be missing, or an error in the question/options/key. However, based on standard interpretation, the answer is 7. I will use 7 as the correct answer.