The quadratic equationx2 + bx + c = 0 has two roots4a and3a, wherea is an integer. Which of the following is a possible value ofb2 +c ?a)3721b)550c)549d)427e)361Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Understanding the Quadratic Equation
The quadratic equation given is x² + bx + c = 0, with roots 4a and 3a. According to Vieta's formulas, the sum and product of the roots can be expressed as:
- Sum of the roots (4a + 3a) = -b
- Product of the roots (4a * 3a) = c
Calculating b and c
From the above, we can derive:
- Sum of the roots: 7a = -b → b = -7a
- Product of the roots: 12a² = c
Finding b² + c
Now, we need to calculate b² + c:
- b² = (-7a)² = 49a²
- c = 12a²
Thus,
- b² + c = 49a² + 12a² = 61a²
Finding Possible Values of b² + c
Now we need to find possible values of 61a² for integer values of a.
- If a = 1, then 61(1)² = 61
- If a = 2, then 61(2)² = 244
- If a = 3, then 61(3)² = 549
- If a = 4, then 61(4)² = 976
The only value from the options given that matches is 549 when a = 3.
Conclusion
Thus, the possible value of b² + c is:
- Option C: 549
This is the correct answer as verified by the calculations based on the properties of the roots in the quadratic equation.

Answer

Given that the quadratic equation x² + bx + c = 0 has roots 4a and 3a, we can use the sum and product of roots formulas to relate these values to the coefficients:

Sum of roots = -b/a = 4a + 3a = 7a
Product of roots = c/a = (4a)(3a) = 12a²

We know that the sum of roots is equal to -b/a, so we have:
-b/a = 7a

From this equation, we can deduce that b = -7a².

Now, let's find the value of b² + c:
b² + c = (-7a²)² + c
b² + c = 49a⁴ + c

Since we are given that a is an integer, let's substitute some values for a and evaluate the expression 49a⁴ + c:

For a = 1:
49(1)⁴ + c = 49 + c

For a = 2:
49(2)⁴ + c = 784 + c

For a = 3:
49(3)⁴ + c = 6561 + c

From the options, only option C (549) can be expressed as 49a⁴ + c, where a is an integer.

Therefore, the answer is C.

Answer

Given that the quadratic equation x² + bx + c = 0 has roots 4a and 3a, we can use the sum and product of roots formulas to relate these values to the coefficients:

Sum of roots = -b/a = 4a + 3a = 7a
Product of roots = c/a = (4a)(3a) = 12a²

We know that the sum of roots is equal to -b/a, so we have:
-b/a = 7a

From this equation, we can deduce that b = -7a².

Now, let's find the value of b² + c:
b² + c = (-7a²)² + c
b² + c = 49a⁴ + c

Since we are given that a is an integer, let's substitute some values for a and evaluate the expression 49a⁴ + c:

For a = 1:
49(1)⁴ + c = 49 + c

For a = 2:
49(2)⁴ + c = 784 + c

For a = 3:
49(3)⁴ + c = 6561 + c

From the options, only option C (549) can be expressed as 49a⁴ + c, where a is an integer.

Therefore, the answer is C.

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