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Which of the following represents all the possible values of x that are solutions to the equation 3x = |x2 -10| ?
- a)-5, -2, and 0
- b)-5, -2 , 2, and 5
- c)-5 and 2
- d)-2 and 5
- e)2 and 5
Correct answer is option 'E'. Can you explain this answer?
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Which of the following represents all the possible values of x that ar...
To solve the equation 3x = |x^2 - 10|, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: x^2 - 10 ≥ 0
This means that x^2 ≥ 10, which implies x ≥ √10 or x ≤ -√10. Since we are looking for integer solutions, we can narrow down the possible values of x to -3, -2, -1, 0, 1, 2, 3, 4, and so on. However, we need to check which values satisfy the equation.
Substituting x = -3, -2, -1, 0, 1, 2, and 3 into 3x = |x^2 - 10|, we find that none of these values satisfy the equation.
Case 2: x^2 - 10 < />
This means that x^2 < 10,="" which="" implies="" -√10="" />< x="" />< √10.="" again,="" since="" we="" are="" looking="" for="" integer="" solutions,="" we="" narrow="" down="" the="" possible="" values="" of="" x="" to="" -3,="" -2,="" -1,="" 0,="" 1,="" 2,="" and="" 3.="" we="" substitute="" these="" values="" into="" the="" />
Substituting x = -3, -2, -1, 0, 1, 2, and 3 into 3x = |x^2 - 10|, we find that x = 2 and x = 5 satisfy the equation.
Therefore, the possible values of x that are solutions to the equation 3x = |x^2 - 10| are 2 and 5.
Hence, the correct answer is option E) 2 and 5.
Case 1: x^2 - 10 ≥ 0
This means that x^2 ≥ 10, which implies x ≥ √10 or x ≤ -√10. Since we are looking for integer solutions, we can narrow down the possible values of x to -3, -2, -1, 0, 1, 2, 3, 4, and so on. However, we need to check which values satisfy the equation.
Substituting x = -3, -2, -1, 0, 1, 2, and 3 into 3x = |x^2 - 10|, we find that none of these values satisfy the equation.
Case 2: x^2 - 10 < />
This means that x^2 < 10,="" which="" implies="" -√10="" />< x="" />< √10.="" again,="" since="" we="" are="" looking="" for="" integer="" solutions,="" we="" narrow="" down="" the="" possible="" values="" of="" x="" to="" -3,="" -2,="" -1,="" 0,="" 1,="" 2,="" and="" 3.="" we="" substitute="" these="" values="" into="" the="" />
Substituting x = -3, -2, -1, 0, 1, 2, and 3 into 3x = |x^2 - 10|, we find that x = 2 and x = 5 satisfy the equation.
Therefore, the possible values of x that are solutions to the equation 3x = |x^2 - 10| are 2 and 5.
Hence, the correct answer is option E) 2 and 5.
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Which of the following represents all the possible values of x that ar...
To find the solutions to the equation 3x = |x2 - 10|, we need to consider two cases:
Case 1: x2 - 10 is non-negative (x2 - 10 ≥ 0):
In this case, the absolute value |x2 - 10| can be simplified to (x2 - 10). Therefore, our equation becomes 3x = x2 - 10.
In this case, the absolute value |x2 - 10| can be simplified to (x2 - 10). Therefore, our equation becomes 3x = x2 - 10.
Rearranging the terms, we have:
x2 - 3x - 10 = 0
x2 - 3x - 10 = 0
Factoring the quadratic equation, we get:
(x - 5)(x + 2) = 0
(x - 5)(x + 2) = 0
Setting each factor equal to zero, we have:
x - 5 = 0 → x = 5
x + 2 = 0 → x = -2
x - 5 = 0 → x = 5
x + 2 = 0 → x = -2
So, in this case, the possible values of x are 5 and -2.
Case 2: x2 - 10 is negative (x2 - 10 < 0):
In this case, the absolute value |x2 - 10| can be simplified to -(x2 - 10), changing the sign of the equation. Therefore, our equation becomes 3x = -(x2 - 10).
In this case, the absolute value |x2 - 10| can be simplified to -(x2 - 10), changing the sign of the equation. Therefore, our equation becomes 3x = -(x2 - 10).
Rearranging the terms, we have:
-3x = x2 - 10
-3x = x2 - 10
Rearranging further, we get:
x2 + 3x - 10 = 0
x2 + 3x - 10 = 0
Factoring the quadratic equation, we have:
(x - 2)(x + 5) = 0
(x - 2)(x + 5) = 0
Setting each factor equal to zero, we have:
x - 2 = 0 → x = 2
x + 5 = 0 → x = -5
x - 2 = 0 → x = 2
x + 5 = 0 → x = -5
So, in this case, the possible values of x are 2 and -5.
Combining the solutions from both cases, we find that the possible values of x that satisfy the equation 3x = |x2 - 10| are -5, -2, 2, and 5.
Therefore, the correct answer is option E: 2 and 5.
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Which of the following represents all the possible values of x that are solutions to the equation 3x = |x2 -10| ?a)-5, -2, and 0b)-5, -2 , 2, and 5c)-5 and 2d)-2 and 5e)2 and 5Correct answer is option 'E'. Can you explain this answer?
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