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A and B solved a quadratic equation. While solving it, A made a mistake in the constant term and obtained the roots as 5, - 3, while B made a mistake in the coefficient of x and obtained the roots as 1, - 3. The correct roots of the equation are
- a)+ 1, + 3
- b)- 1, 3
- c)- 1, - 3
- d)1, - 1
Correct answer is option 'B'. Can you explain this answer?
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A and B solved a quadratic equation. While solving it, A made a mista...
Given Information:
A and B solved a quadratic equation. A made a mistake in the constant term and obtained the roots as 5 and -3. B made a mistake in the coefficient of x and obtained the roots as 1 and -3.
Approach:
To find the correct roots of the quadratic equation, we need to identify the mistakes made by A and B and then correct them.
Solution:
Let's assume the quadratic equation to be ax^2 + bx + c = 0.
Mistake made by A:
A obtained the roots as 5 and -3. This means that the equation with the constant term mistake can be written as:
a(x - 5)(x + 3) = 0
Expanding the equation, we get:
a(x^2 - 2x - 15) = 0
ax^2 - 2ax - 15a = 0
From this equation, we can see that the constant term is -15a. The correct constant term should be c/a. So, we have the equation:
-15a = c/a
Simplifying this equation, we get:
c = -15
Mistake made by B:
B obtained the roots as 1 and -3. This means that the equation with the coefficient mistake can be written as:
(x - 1)(x + 3) = 0
Expanding the equation, we get:
x^2 + 2x - 3 = 0
From this equation, we can see that the coefficient of x is 2. The correct coefficient of x should be b/a. So, we have the equation:
2 = b/a
Simplifying this equation, we get:
b = 2a
Correcting the Mistakes:
Now that we have identified the mistakes made by A and B, we can correct them.
From the mistake made by A, we found that c = -15.
From the mistake made by B, we found that b = 2a.
Substituting these values in the quadratic equation ax^2 + bx + c = 0, we get:
ax^2 + (2a)x - 15 = 0
Now, we can solve this quadratic equation to find its roots.
Using the Quadratic Formula:
The quadratic formula states that for an equation ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Substituting the values of a, b, and c in the formula, we get:
x = (-(2a) ± √((2a)^2 - 4a(-15)))/(2a)
Simplifying this equation, we get:
x = (-2a ± √(4a^2 + 60a))/(2a)
x = (-2a ± √(4a(a + 15)))/(2a)
x = (-2a ± 2√(a(a + 15)))/(2a)
x = -1 ± √(a(a + 15))/a
From this equation, we can see that the roots are -1 ± √(a(a + 15
A and B solved a quadratic equation. A made a mistake in the constant term and obtained the roots as 5 and -3. B made a mistake in the coefficient of x and obtained the roots as 1 and -3.
Approach:
To find the correct roots of the quadratic equation, we need to identify the mistakes made by A and B and then correct them.
Solution:
Let's assume the quadratic equation to be ax^2 + bx + c = 0.
Mistake made by A:
A obtained the roots as 5 and -3. This means that the equation with the constant term mistake can be written as:
a(x - 5)(x + 3) = 0
Expanding the equation, we get:
a(x^2 - 2x - 15) = 0
ax^2 - 2ax - 15a = 0
From this equation, we can see that the constant term is -15a. The correct constant term should be c/a. So, we have the equation:
-15a = c/a
Simplifying this equation, we get:
c = -15
Mistake made by B:
B obtained the roots as 1 and -3. This means that the equation with the coefficient mistake can be written as:
(x - 1)(x + 3) = 0
Expanding the equation, we get:
x^2 + 2x - 3 = 0
From this equation, we can see that the coefficient of x is 2. The correct coefficient of x should be b/a. So, we have the equation:
2 = b/a
Simplifying this equation, we get:
b = 2a
Correcting the Mistakes:
Now that we have identified the mistakes made by A and B, we can correct them.
From the mistake made by A, we found that c = -15.
From the mistake made by B, we found that b = 2a.
Substituting these values in the quadratic equation ax^2 + bx + c = 0, we get:
ax^2 + (2a)x - 15 = 0
Now, we can solve this quadratic equation to find its roots.
Using the Quadratic Formula:
The quadratic formula states that for an equation ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Substituting the values of a, b, and c in the formula, we get:
x = (-(2a) ± √((2a)^2 - 4a(-15)))/(2a)
Simplifying this equation, we get:
x = (-2a ± √(4a^2 + 60a))/(2a)
x = (-2a ± √(4a(a + 15)))/(2a)
x = (-2a ± 2√(a(a + 15)))/(2a)
x = -1 ± √(a(a + 15))/a
From this equation, we can see that the roots are -1 ± √(a(a + 15
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Community Answer
A and B solved a quadratic equation. While solving it, A made a mista...
A made a mistake in the constant term and obtained the roots 5, - 3.
Therefore, the sum of roots = 5 - 3 = 2
B made a mistake in the coefficient of x and obtained the roots 1, - 3.
Therefore, the product of roots = 1 × - 3 = - 3
There is only one option whose sum is 2 and product is - 3, i.e. option (2).
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