SOLUTION: solve equation by completing the square z^2-5z-7=0

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Question 133289: solve equation by completing the square
z^2-5z-7=0
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1+z%5E2-5+z-7=0 Start with the given expression


1%28z%5E2-5z-7%29=0 Factor out the leading coefficient 1


Take half of the z coefficient -5 to get -5%2F2 (ie %281%2F2%29%28-5%29=-5%2F2).

Now square -5%2F2 to get 25%2F4 (ie %28-5%2F2%29%5E2=%28-5%2F2%29%28-5%2F2%29=25%2F4)




1%28z%5E2-5z%2B25%2F4-25%2F4-7%29=0 Now add and subtract this value inside the parenthesis. Notice how 25%2F4-25%2F4=0. Since we're adding 0, we're not changing the equation.



1%28%28z-5%2F2%29%5E2-25%2F4-7%29=0 Now factor z%5E2-5z%2B25%2F4 to get %28z-5%2F2%29%5E2



1%28%28z-5%2F2%29%5E2-53%2F4%29=0 Combine like terms



1%28z-5%2F2%29%5E2%2B1%28-53%2F4%29=0 Distribute



1%28z-5%2F2%29%5E2-53%2F4=0 Multiply




1%28z-5%2F2%29%5E2=53%2F4 Add 53%2F4 to both sides




%28z-5%2F2%29%5E2=%2853%2F4%29%2F%281%29 Divide both sides by 1



%28z-5%2F2%29%5E2=53%2F4 Reduce



Take the square root of both sides



Simplify



Add 5%2F2 to both sides



Combine the fractions



Break up the expression


or



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Answer:


So our solutions are


or

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
z%5E2-5z-7=0


Step 1: Move the constant term to the right hand side of the equation:

z%5E2-5z=7

Step 2: Divide the coefficient of the 1st order term by 2, then square the result:

%28%28-5%29%2F2%29%5E2=25%2F4

Step 3: Add the result of step 2 to both sides of the equation:

z%5E2-5z%2B25%2F4=7%2B25%2F4=28%2F4%2B25%2F4=53%2F4

Step 4: The result of step 3 was that you have a perfect square on the left, so factor it:

%28z-5%2F2%29%5E2=53%2F4

Step 5: Take the square root of both sides, remembering to consider both the positive and negative roots:

z-5%2F2=sqrt%2853%29%2F2 or z-5%2F2=-sqrt%2853%29%2F2

Step 6: Move the constant term on the left to the right:

z=%285%2Bsqrt%2853%29%29%2F2 or z=%285-sqrt%2853%29%29%2F2

Step 7: Simplify if possible. In this case, there is no simplification possible because 53 is prime.