A number increased by 10 is 114
2 answers:
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Answer:
n = 6 + or n = 6 -
Step-by-step explanation:
We can solve this equation using the quadratic formula OR Completing the Square method.
n² + 14 = 12n
rearrange : n² - 12n + 14 = 0
here a= 1 , b = -12, c = 14
the quadratic formula says: x = - b/ (2a) + root(b^2 - 4ac) / (2a)
or x = - b/ (2a) - root(b^2 - 4ac) / (2a)
x = - (-12)/ (2) + root((-12)^2 - 4*14) / (2)
x = 6 + root (144 - 56) / 2
x = 6 + root(88)/2
x = 6 + root(4*22) / 2
x = 6 + 2*root(22)/2
x = 6 + root(22) = 6 +
so x =6 + or x = 6 -
In this case x = n
n = 6 + or n = 6 -
Given the points P(-2, 7), Q(6, 8) and R(-1, -3), we find the quadratic equation of the form 
as follows:
Step 1:
We check to make sure that no pair of points is colinear (i.e. lie on the same line) by making that the slopes of the combinations of the line are not equal.
Slope of PQ =
Slope of PR =
Slope of QR =
Since, the slope of the lines are not equal, thus the points are not colinear nor parallel.
Step 2:
Substitute the x-values and the y-values of the given points into the quadratic equation formular to have three system of equations as follows:
Step 3:
We solve the system of equations as follows:
![\left[\begin{array}{ccccc}4&-2&1&|&7\\36&6&1&|&8\\1&-1&1&|&-3\end{array}\right] \ \ \ R_1\leftrightarrow R_3 \\ \\ \left[\begin{array}{ccccc}1&-1&1&|&-3\\36&6&1&|&8\\4&-2&1&|&7\end{array}\right] \ \ \ {{-36R_1+R_2\rightarrow R_2} \atop {-4R_1+R_3\rightarrow R_3}} \\ \\ \left[\begin{array}{ccccc}1&-1&1&|&-3\\0&42&-35&|&116\\0&2&-3&|&19\end{array}\right] \ \ \ R_2\leftrightarrow R_3](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D4%26-2%261%26%7C%267%5C%5C36%266%261%26%7C%268%5C%5C1%26-1%261%26%7C%26-3%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20R_1%5Cleftrightarrow%20R_3%20%5C%5C%20%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%26-1%261%26%7C%26-3%5C%5C36%266%261%26%7C%268%5C%5C4%26-2%261%26%7C%267%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20%7B%7B-36R_1%2BR_2%5Crightarrow%20R_2%7D%20%5Catop%20%7B-4R_1%2BR_3%5Crightarrow%20R_3%7D%7D%20%5C%5C%20%20%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%26-1%261%26%7C%26-3%5C%5C0%2642%26-35%26%7C%26116%5C%5C0%262%26-3%26%7C%2619%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20R_2%5Cleftrightarrow%20R_3)
![\left[\begin{array}{ccccc}1&-1&1&|&-3\\0&2&-3&|&19\\0&42&-35&|&116\end{array}\right] \ \ \ \frac{1}{2} R_2\leftrightarrow R_2 \\ \\ \left[\begin{array}{ccccc}1&-1&1&|&-3\\0&1&\frac{-3}{2}&|&\frac{19}{2}\\0&42&-35&|&116\end{array}\right] \ \ \ {{R_1+R_2\rightarrow R_1} \atop {-42R_2+R_3\rightarrow R_3}} \\ \\ \left[\begin{array}{ccccc}1&0&\frac{-1}{2}&|&\frac{13}{2}\\0&1&\frac{-3}{2}&|&\frac{19}{2}\\0&0&28&|&-283\end{array}\right] \ \ \ \frac{1}{28} R_3\leftrightarrow R_3](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%26-1%261%26%7C%26-3%5C%5C0%262%26-3%26%7C%2619%5C%5C0%2642%26-35%26%7C%26116%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20%20%5Cfrac%7B1%7D%7B2%7D%20R_2%5Cleftrightarrow%20R_2%20%5C%5C%20%20%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%26-1%261%26%7C%26-3%5C%5C0%261%26%5Cfrac%7B-3%7D%7B2%7D%26%7C%26%5Cfrac%7B19%7D%7B2%7D%5C%5C0%2642%26-35%26%7C%26116%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20%20%7B%7BR_1%2BR_2%5Crightarrow%20R_1%7D%20%5Catop%20%7B-42R_2%2BR_3%5Crightarrow%20R_3%7D%7D%20%5C%5C%20%20%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%260%26%5Cfrac%7B-1%7D%7B2%7D%26%7C%26%5Cfrac%7B13%7D%7B2%7D%5C%5C0%261%26%5Cfrac%7B-3%7D%7B2%7D%26%7C%26%5Cfrac%7B19%7D%7B2%7D%5C%5C0%260%2628%26%7C%26-283%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20%5Cfrac%7B1%7D%7B28%7D%20R_3%5Cleftrightarrow%20R_3)
![\left[\begin{array}{ccccc}1&0&\frac{-1}{2}&|&\frac{13}{2}\\0&1&\frac{-3}{2}&|&\frac{19}{2}\\0&0&1&|&\frac{-283}{28}\end{array}\right] \ \ \ {{\frac{1}{2}R_3+R_1\rightarrow R_1} \atop {\frac{3}{2}R_3+R_2\rightarrow R_2}} \\ \\ \left[\begin{array}{ccccc}1&0&0&|&\frac{81}{56}\\0&1&0&|&\frac{-317}{56}\\0&0&1&|&\frac{-283}{28}\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%260%26%5Cfrac%7B-1%7D%7B2%7D%26%7C%26%5Cfrac%7B13%7D%7B2%7D%5C%5C0%261%26%5Cfrac%7B-3%7D%7B2%7D%26%7C%26%5Cfrac%7B19%7D%7B2%7D%5C%5C0%260%261%26%7C%26%5Cfrac%7B-283%7D%7B28%7D%5Cend%7Barray%7D%5Cright%5D%20%5C%20%5C%20%5C%20%7B%7B%5Cfrac%7B1%7D%7B2%7DR_3%2BR_1%5Crightarrow%20R_1%7D%20%5Catop%20%7B%5Cfrac%7B3%7D%7B2%7DR_3%2BR_2%5Crightarrow%20R_2%7D%7D%20%5C%5C%20%20%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%260%260%26%7C%26%5Cfrac%7B81%7D%7B56%7D%5C%5C0%261%260%26%7C%26%5Cfrac%7B-317%7D%7B56%7D%5C%5C0%260%261%26%7C%26%5Cfrac%7B-283%7D%7B28%7D%5Cend%7Barray%7D%5Cright%5D)
Thus,
Therefore, the quadratic equation with the three given points is
Step 1:
We check to make sure that no pair of points is colinear (i.e. lie on the same line) by making that the slopes of the combinations of the line are not equal.
Slope of PQ =
Slope of PR =
Slope of QR =
Since, the slope of the lines are not equal, thus the points are not colinear nor parallel.
Step 2:
Substitute the x-values and the y-values of the given points into the quadratic equation formular to have three system of equations as follows:
Step 3:
We solve the system of equations as follows:
Thus,
Therefore, the quadratic equation with the three given points is
The cost of the wood will be $1,003.05
<h3>Volume of a cylinder </h3>The formula for calculating the volume of a cylinder is expressed as:
where:
- r is the radius
- h is the height
Given the following parameters
r = d/2 = 1.5/2 = 0.75ft
h = 45ft
Get the volume of the cylinder
If the <u>wood costs $12.62 per cubic foot,</u> hence the cost of the wood will be 79.4815 * 12.62 = $1,003.05
Learn more on volume of cylinder here: brainly.com/question/9554871