Are these right?!
Absolute value :
1. |x^2-2x+16|=8
x=-4
x=-3
x=-2
x=2
2.|3x-2|=7
x=5\3
x=3
3. |3x+2|+3=0
x= -5/3
1.
x^2 - 2x + 16 = 8 or x^2 - 2x + 16 = -8
x^2 - 2x + 8 = 0 or x^2 - 2x + 24 = 0
neither the first nor the second has a real solution.
2. |3x-2| = 7
3x-2=7 or 3x-2=-7
3x=9 or 3x = -5
x = 3 or x = -5/3
3.
|3x+2| = -3
by its very definition, there cannot be a solution here.
In the first problem it was suppose to be -16
my mistake, but does it still g=have no solution?
To check if the solutions you provided are correct for the given absolute value equations, let's solve each equation step by step:
1. |x^2-2x+16|=8
To solve this equation, we set both sides equal to positive and negative the given value:
Positive case: x^2-2x+16 = 8
Rearrange the equation: x^2 - 2x + 8 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -2, and c = 8:
x = (-(-2) ± √((-2)^2 - 4(1)(8))) / (2(1))
x = (2 ± √(4 - 32)) / 2
x = (2 ± √(-28)) / 2
Since we have a square root of a negative number, there are no real solutions in this case.
Negative case: -(x^2-2x+16) = 8
Rearrange the equation: -x^2 + 2x - 16 = 8
To solve this quadratic equation, we can rearrange it and solve:
x^2 - 2x - 24 = 0
Factoring this quadratic equation, we get:
(x - 6)(x + 4) = 0
Setting each factor equal to zero, we have two possible solutions:
x - 6 = 0 -> x = 6
x + 4 = 0 -> x = -4
Therefore, the correct solutions for the equation |x^2-2x+16|=8 are x = 6 and x = -4. So, the solutions you provided, x = -3, x = -2, and x = 2, are not correct.
2. |3x-2|=7
To solve this equation, we set both sides equal to positive and negative the given value:
Positive case: 3x - 2 = 7
Solve for x: 3x = 9 -> x = 3
Negative case: -(3x-2) = 7
Solve for x: -3x + 2 = 7 -> -3x = 5 -> x = -5/3
Therefore, the correct solutions for the equation |3x-2|=7 are x = 3 and x = -5/3. So, the solution you provided, x = 5/3, is not correct.
3. |3x+2|+3 = 0
To solve this equation, let's isolate the absolute value expression:
|3x + 2| = -3
Absolute values cannot be negative, so there are no solutions in this case. The absolute value of any expression is always non-negative.
Therefore, the equation |3x+2|+3 = 0 has no real solutions. So, the solution you provided, x = -5/3, is not correct.
Overall, the correct solutions for the given absolute value equations are:
1. |x^2-2x+16|=8 -> x = 6 and x = -4
2. |3x-2|=7 -> x = 3 and x = -5/3
3. |3x+2|+3 = 0 -> No real solutions