SOLUTION: Three parts and uses the quadratic equation: x^2 - 18x + 72 = 0 Part a. 1. Graph the quadratic equation and: a. Label and include the ordered pair for the vertex on the grap

Algebra ->  Systems-of-equations -> SOLUTION: Three parts and uses the quadratic equation: x^2 - 18x + 72 = 0 Part a. 1. Graph the quadratic equation and: a. Label and include the ordered pair for the vertex on the grap      Log On


   

Question 900370: Three parts and uses the quadratic equation: x^2 - 18x + 72 = 0
Part a.
1. Graph the quadratic equation and:
a. Label and include the ordered pair for the vertex on the graph.
b. Label and include the ordered pair(s) for the x--intercepts on the graph.
c. How are the solutions identified from the graph?
Part b.
1. Solve the quadratic equation by using the "completing the square" method.
2. What do you notice about the answer you found here and where the x-intercepts of the graph are in part a?
Part c.
1. Solve the quadratic equation by using the quadratic formula. Show all your work.
2. Compare your answer to what you found in part a and b. What do you notice about the answers?

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics
Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert 1x%5E2%2B-18x%2B72=0 to standard form by dividing both sides by 1:
We have: 1x%5E2%2B-18x%2B72=0. What we want to do now is to change this equation to a complete square %28x%2Bsomenumber%29%5E2+%2B+othernumber. How can we find out values of somenumber and othernumber that would make it work?
Look at %28x%2Bsomenumber%29%5E2: %28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2. Since the coefficient in our equation 1x%5E2%2Bhighlight_red%28+-18%29+%2A+x%2B72=0 that goes in front of x is -18, we know that -18=2*somenumber, or somenumber+=+-18%2F2. So, we know that our equation can be rewritten as %28x%2B-18%2F2%29%5E2+%2B+othernumber, and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that %28x%2B-18%2F2%29%5E2+%2B+othernumber is equivalent to our original equation 1x%5E2%2B-18x%2Bhighlight_green%28+72+%29=0.


The highlighted red part must be equal to 72 (highlighted green part).

-18%5E2%2F4+%2B+othernumber+=+72, or othernumber+=+72--18%5E2%2F4+=+-9.
So, the equation converts to %28x%2B-18%2F2%29%5E2+%2B+-9+=+0, or %28x%2B-18%2F2%29%5E2+=+9.

Our equation converted to a square %28x%2B-18%2F2%29%5E2, equated to a number (9).

Since the right part 9 is greater than zero, there are two solutions:

system%28+%28x%2B-18%2F2%29+=+%2Bsqrt%28+9+%29%2C+%28x%2B-18%2F2%29+=+-sqrt%28+9+%29+%29
, or

system%28+%28x%2B-18%2F2%29+=+3%2C+%28x%2B-18%2F2%29+=+-3+%29
system%28+x%2B-18%2F2+=+3%2C+x%2B-18%2F2+=+-3+%29
system%28+x+=+3--18%2F2%2C+x+=+-3--18%2F2+%29

system%28+x+=+12%2C+x+=+6+%29
Answer: x=12, 6.

Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2-18%2Ax%2B72=0 ( notice a=1, b=-18, and c=72)





x+=+%28--18+%2B-+sqrt%28+%28-18%29%5E2-4%2A1%2A72+%29%29%2F%282%2A1%29 Plug in a=1, b=-18, and c=72




x+=+%2818+%2B-+sqrt%28+%28-18%29%5E2-4%2A1%2A72+%29%29%2F%282%2A1%29 Negate -18 to get 18




x+=+%2818+%2B-+sqrt%28+324-4%2A1%2A72+%29%29%2F%282%2A1%29 Square -18 to get 324 (note: remember when you square -18, you must square the negative as well. This is because %28-18%29%5E2=-18%2A-18=324.)




x+=+%2818+%2B-+sqrt%28+324%2B-288+%29%29%2F%282%2A1%29 Multiply -4%2A72%2A1 to get -288




x+=+%2818+%2B-+sqrt%28+36+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%2818+%2B-+6%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%2818+%2B-+6%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%2818+%2B+6%29%2F2 or x+=+%2818+-+6%29%2F2


Lets look at the first part:


x=%2818+%2B+6%29%2F2


x=24%2F2 Add the terms in the numerator

x=12 Divide


So one answer is

x=12




Now lets look at the second part:


x=%2818+-+6%29%2F2


x=12%2F2 Subtract the terms in the numerator

x=6 Divide


So another answer is

x=6


So our solutions are:

x=12 or x=6