SOLUTION: c..ex... Problem #22 Solve -9(x-3)^2 = -7 Problem #33 Solve by completeing the square 2x^2-4x-11=0

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: c..ex... Problem #22 Solve -9(x-3)^2 = -7 Problem #33 Solve by completeing the square 2x^2-4x-11=0      Log On


   

Question 77972: c..ex...
Problem #22
Solve
-9(x-3)^2 = -7
Problem #33
Solve by completeing the square
2x^2-4x-11=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#22
-9%28x-3%29%5E2+=+-7
%28x-3%29%5E2+=+7%2F9 Divide both sides by -9
x-3+=0%2B-+sqrt%287%2F9%29 Take the square root of both sides
x-3+=0%2B-+sqrt%287%29%2F3 Reduce the denominator
x+=0%2B-+sqrt%287%29%2F3%2B3 Add 3 to both sides so our solution is


#33
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=2+x%5E2-4+x-11 Start with the given equation



y%2B11=2+x%5E2-4+x Add 11 to both sides



y%2B11=2%28x%5E2-2x%29 Factor out the leading coefficient 2



Take half of the x coefficient -2 to get -1 (ie %281%2F2%29%28-2%29=-1).


Now square -1 to get 1 (ie %28-1%29%5E2=%28-1%29%28-1%29=1)





y%2B11=2%28x%5E2-2x%2B1-1%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1 does not change the equation




y%2B11=2%28%28x-1%29%5E2-1%29 Now factor x%5E2-2x%2B1 to get %28x-1%29%5E2



y%2B11=2%28x-1%29%5E2-2%281%29 Distribute



y%2B11=2%28x-1%29%5E2-2 Multiply



y=2%28x-1%29%5E2-2-11 Now add %2B11 to both sides to isolate y



y=2%28x-1%29%5E2-13 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=2, h=1, and k=-13. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=2x%5E2-4x-11 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x%5E2-4x-11%29 Graph of y=2x%5E2-4x-11. Notice how the vertex is (1,-13).



Notice if we graph the final equation y=2%28x-1%29%5E2-13 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%28x-1%29%5E2-13%29 Graph of y=2%28x-1%29%5E2-13. Notice how the vertex is also (1,-13).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.





So we now have
0=2%28x-1%29%5E2-13
13=2%28x-1%29%5E2 Add 13 to both sides
13%2F2=%28x-1%29%5E2 Divide both sides by 2
0%2B-sqrt%2813%2F2%29=x-1 Take the square root of both sides
0%2B-sqrt%2813%2F2%29%2B1=x Add 1 to both sides
So our answer is
x=0%2B-sqrt%2813%2F2%29%2B1