CAT Question > A quadratic equation f(x) = 0 exists such tha...
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A quadratic equation f(x) = 0 exists such that f(x) = 3x2 + 9x + 3. A second quadratic expression g(x) is formed by shifting f(x) towards the right along the x-axis by 5 units. What is the sum of the roots of equation g(x) = 0?
- a)7
- b)10
- c)21
- d)-7
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
A quadratic equation f(x) = 0 exists such that f(x) = 3x2 + 9x + 3. A ...
g(x) is formed by shifting f(x) towards the right along the x-axis by 5 units,
g(x) = 3(x−5)2 + 9(x−5) + 3
g(x) = 3(x2−10x+25) + 9(x−5) + 3
g(x) = 3x2 − 30x + 75 + 9x − 45 + 3
g(x) = 3x2 − 21x + 33
Hence g(x) = 0
3x2 − 21x + 33 = 0
x2 − 7x + 11 = 0
Hence, sum of the roots = (−7)/1 = 7
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A quadratic equation f(x) = 0 exists such that f(x) = 3x2 + 9x + 3. A ...
Given quadratic equation:
f(x) = 3x^2 - 9x + 3
Shifting f(x) to the right:
To shift the function f(x) to the right by 5 units, we substitute x with (x-5) in the equation.
Forming the shifted quadratic equation:
g(x) = 3(x-5)^2 - 9(x-5) + 3
= 3(x^2 - 10x + 25) - 9x + 45 + 3
= 3x^2 - 30x + 75 - 9x + 45 + 3
= 3x^2 - 39x + 123
Finding the roots of g(x):
To find the roots of g(x), we set g(x) equal to zero and solve for x.
3x^2 - 39x + 123 = 0
Using the quadratic formula:
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
For our equation, a = 3, b = -39, and c = 123.
x = (-(-39) ± √((-39)^2 - 4(3)(123))) / (2(3))
= (39 ± √(1521 - 1476)) / 6
= (39 ± √45) / 6
Simplifying the expression:
x = (39 + √45) / 6 or x = (39 - √45) / 6
Sum of the roots:
The sum of the roots of a quadratic equation can be found by adding the roots.
Sum of roots = (39 + √45) / 6 + (39 - √45) / 6
= (39 + √45 + 39 - √45) / 6
= (78) / 6
= 13
Therefore, the sum of the roots of the equation g(x) = 0 is 13.
f(x) = 3x^2 - 9x + 3
Shifting f(x) to the right:
To shift the function f(x) to the right by 5 units, we substitute x with (x-5) in the equation.
Forming the shifted quadratic equation:
g(x) = 3(x-5)^2 - 9(x-5) + 3
= 3(x^2 - 10x + 25) - 9x + 45 + 3
= 3x^2 - 30x + 75 - 9x + 45 + 3
= 3x^2 - 39x + 123
Finding the roots of g(x):
To find the roots of g(x), we set g(x) equal to zero and solve for x.
3x^2 - 39x + 123 = 0
Using the quadratic formula:
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
For our equation, a = 3, b = -39, and c = 123.
x = (-(-39) ± √((-39)^2 - 4(3)(123))) / (2(3))
= (39 ± √(1521 - 1476)) / 6
= (39 ± √45) / 6
Simplifying the expression:
x = (39 + √45) / 6 or x = (39 - √45) / 6
Sum of the roots:
The sum of the roots of a quadratic equation can be found by adding the roots.
Sum of roots = (39 + √45) / 6 + (39 - √45) / 6
= (39 + √45 + 39 - √45) / 6
= (78) / 6
= 13
Therefore, the sum of the roots of the equation g(x) = 0 is 13.
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A quadratic equation f(x) = 0 exists such that f(x) = 3x2 + 9x + 3. A second quadratic expression g(x) is formed by shifting f(x) towards the right along the x-axis by 5 units. What is the sum of the roots of equation g(x) = 0?a)7b)10c)21d)-7Correct answer is option 'A'. Can you explain this answer?
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