Question

1. Solve for x in the equation below.

(x-4)^2+3=12
A. X=7
B. X=-1,1
C. 1,7
D. 7,7
2. Solve for y in the equation below. -6y^2-96y-360=0
A. Y=-10,6
B. Y=-10,
C. -6,10
D. 6,10
3. Solve for x in the equation below.
X^2+2x-3=-2x-8
A. X=-2-i-,-2+i
B. No solutions
C. X=2-i,2+i
D. X=5,-1
4. Solve for z in the inequality below.
z^2−z−6≥0
A. z≤−2 or z≥3
B. z≤−3 or z≥2
C. −2≤z≤3
D. −3≤z≤2

Answer 1

[AI-generated answer — Bot GPT 3.5]

1. To solve for x, we first need to simplify the equation:

(x-4)^2 + 3 = 12
Expanding the square:
(x^2 - 8x + 16) + 3 = 12
Combining like terms:
x^2 - 8x + 19 = 12
Moving all terms to one side:
x^2 - 8x + 7 = 0
This is a quadratic equation, so we can solve by factoring, using the quadratic formula, or by completing the square. In this case, we can factor it:
(x - 7)(x - 1) = 0
Setting each factor equal to zero:
x - 7 = 0 or x - 1 = 0
Solving for x:
x = 7 or x = 1

Therefore, the solution is x = 7 or x = 1.

A. X = 7
C. 1, 7 (Correct)

2. To solve for y in the equation -6y^2-96y-360 = 0, we can first divide all terms by -6 to simplify the equation:
y^2 + 16y + 60 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula to solve for y:
y = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 1, b = 16, and c = 60 into the formula:
y = (-16 ± √(16^2 - 4(1)(60)))/(2(1))
Simplifying:
y = (-16 ± √(256 - 240))/(2)
y = (-16 ± √16)/(2)
y = (-16 ± 4)/(2)
This gives two possible solutions:
y = (-16 + 4)/(2) = -12/2 = -6
y = (-16 - 4)/(2) = -20/2 = -10

Therefore, the solution is y = -10 or y = -6.

A. Y = -10, 6
B. Y = -10, (Incorrect)
D. 6, 10 (Incorrect)

3. To solve for x in the equation x^2 + 2x - 3 = -2x - 8, we can start by combining like terms:
x^2 + 2x + 2x - 3 = -2x - 8
Simplifying further:
x^2 + 4x - 3 = -2x - 8
Moving all terms to one side:
x^2 + 4x + 2x + 3 + 8 = 0
x^2 + 6x + 11 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 1, b = 6, and c = 11 into the formula:
x = (-6 ± √(6^2 - 4(1)(11)))/(2(1))
x = (-6 ± √(36 - 44))/(2)
x = (-6 ± √(-8))/(2)
Since we have a square root of a negative number, there are no real solutions.

Therefore, the answer is:
B. No solutions

4. To solve for z in the inequality z^2 - z - 6 ≥ 0, we can first factor the quadratic:
(z - 3)(z + 2) ≥ 0
Setting each factor greater than or equal to zero:
z - 3 ≥ 0 or z + 2 ≥ 0
Solving for z:
z ≥ 3 or z ≥ -2
Since both factors are greater than or equal to zero, the values of z that satisfy the inequality are:
z ≤ -2 or z ≥ 3

Therefore, the solution is:
B. z ≤ -3 or z ≥ 2