What is the area of the square
1 answer:
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Answer:
1. (x - 3)² = 8
2. (x + 2)² = 3
3. (x + 6)² =
4. (x + 3)² = 27
5. (x + 4)² = 13
6.
Step-by-step explanation:
Completion of Square:
In the following problems the terms in the RHS of the above equation may be missing. We balance the equation. Simplify it and re write it in terms of LHS.
1. x² - 6x + 1 = 0
Taking the constant term to the other side, we get:
x² - 6x = - 1
⇒ x² - 2(3)x = -1
⇒ x² -2(3)x + 9 = - 1 + 9 [Adding 9 to both the sides]
⇒ x² -2(3)x + 3² = 8
⇒ (x - 3)² = 8 is the answer.
2. 3x² + 12x + 3 = 0
Note that the co-effecient of x² is not 1. We make it 1, by dividing the whole equation by 3. And then proceed like the previous problem.
3x² + 12x = -3
Dividing by 3 through out, x² + 4x = - 1
⇒ x² + 2(2) + 4 = -1 + 4
⇒ x² +2(2) + 2² = 3
⇒ (x + 2)² = 3 is the answer.
3. 2x² + 24x = 29
x² + 12x =
⇒ x² + 2(6)x + 36 = + 36
⇒ x² + 2(6)x + 6² =
⇒ (x + 6)² = is the answer.
4. x² + 6x - 18 = 0
x² + 6x = 18
⇒ x² + 2(3)x = 18
⇒ x² + 2(3)x + 9 = 18 + 9
⇒ x² + 2(3)x + 3² = 27
⇒ (x + 3)² = 27 is the answer.
5. x² + 8x + 3 = 0
x² + 8x = -3
⇒ x² + 2(4)x = -3
⇒ x² + 2(4)x + 16 = - 3 + 16
⇒ x² + 2(4)x + 16 = 13
⇒ (x + 4)² = 13 is the answer.
6. 9x² - 30x + 6 = 0
9x² - 30x = - 6
⇒ x² x = - 6
is the answer.
Answer:
a) P(-z_0.025 < Z < z_0.025)
For this case we want a quantile that accumulates 0.025 of the area on the tails of the normal standard distribution, and for this case we can calculate the z value with the following excel codes:
"=NORM.INV(0.025,0,1)"
"=NORM.INV(0.025,0,1)"
And for this case the two values are :
b) P(-z_{\alpha/2} < Z < z_{\alpha/2})
For this case we want a quantile that accumulates of the area on the tails of the normal standard distribution, and for this case we can calculate the z value with the following excel codes:
"=NORM.INV(alpha/2,0,1)"
"=NORM.INV(alpha/2,0,1)"
c) For this case we want to find a value of z that satisfy:
P(Z > z_alpha) = 0.05.
And we can use the following excel code:
"=NORM.INV(0.95,0,1)"
And we got
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Part a
P(-z_0.025 < Z < z_0.025)
For this case we want a quantile that accumulates 0.025 of the area on the tails of the normal standard distribution, and for this case we can calculate the z value with the following excel codes:
"=NORM.INV(0.025,0,1)"
"=NORM.INV(0.025,0,1)"
And for this case the two values are :
Part b
P(-z_{\alpha/2} < Z < z_{\alpha/2})
For this case we want a quantile that accumulates of the area on the tails of the normal standard distribution, and for this case we can calculate the z value with the following excel codes:
"=NORM.INV(alpha/2,0,1)"
"=NORM.INV(alpha/2,0,1)"
Part c
For this case we want to find a value of z that satisfy:
P(Z > z_alpha) = 0.05.
And we can use the following excel code:
"=NORM.INV(0.95,0,1)"
And we got