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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?
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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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Most Upvoted Answer
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.
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Pieces of music, or performances of them, are usually said to be happy, sad, and so on. Musics emotional expressivity is a philosophical problem since the paradigm expressers of emotions are psychological agents, who have emotions to express. Neither pieces of music, nor performances of them, are psychological agents, thus it is puzzling that such things could be said to express emotions. One immediately helpful distinction is that between expression and expressivity, or expressiveness. Expression is something persons do, namely, the outward manifestation of their emotional states. Expressivity is something artworks, and possibly other things, possess. It is presumably related in some way to expression, and yet cannot simply be expression for the reason just given. Most theoristsalso distinguish between expressivity and representation, claiming that music is expressive of emotions, rather than representing them. To give a non-musical example, one might paint a person crying, yet do so in a clinical style such that the painting represents the persons sadness, yet is itself not a sad painting, that is, expressive of sadness, but rather a cool, detached one. The emotions in a piece of music are thus, more closely tied to it than mere descriptions of emotional states.An obvious way to connect expressivity with expression is to argue that pieces of music or performances of them are expressions of emotion - not the pieces or performances emotions, but rather those of the composer or performer. There are two major problems with this ‘expression theory’. The first is that neither composers nor performers often experience the emotions their music is expressive of as it is produced. Nor does it seem unlikely that a composer could create, or a performer perform, a piece expressive of an emotion that she had never experienced. This is not to deny that a composer could write a piece expressive of her emotional state, but two things must be observed. The first is that for the expression theory to be an account of musical expressivity, at least all central cases of expressivity must follow this model, which is not the case. The second is that if a composer is to express her sadness, say, by writing a sad piece, she must write the right kind of piece. In other words, if she is a bad composer she might fail to express her emotion. This brings us to the second major problem for the expression theory. If a composer can fail to express her emotions in a piece, then the music she writes is expressive independently of the emotion she is experiencing. Thus, musics expressivity cannot be explained in terms of direct expression.Q. Which of the following is untrue about a piece of music?

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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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