SOLUTION: Please help show me how to solve the quadratic equation by completing the square z^2 + 8z +9 = 0

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Question 170120: Please help show me how to solve the quadratic equation by completing the square
z^2 + 8z +9 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

z%5E2%2B8z%2B9 Start with the left side of the equation.


Take half of the z coefficient 8 to get 4. In other words, %281%2F2%29%288%29=4.


Now square 4 to get 16. In other words, %284%29%5E2=%284%29%284%29=16


z%5E2%2B8z%2Bhighlight%2816-16%29%2B9 Now add and subtract 16. Make sure to place this after the "z" term. Notice how 16-16=0. So the expression is not changed.


%28z%5E2%2B8z%2B16%29-16%2B9 Group the first three terms.


%28z%2B4%29%5E2-16%2B9 Factor z%5E2%2B8z%2B16 to get %28z%2B4%29%5E2.


%28z%2B4%29%5E2-7 Combine like terms.


So after completing the square, z%5E2%2B8z%2B9 transforms to %28z%2B4%29%5E2-7. So z%5E2%2B8z%2B9=%28z%2B4%29%5E2-7.


So z%5E2%2B8z%2B9=0 is equivalent to %28z%2B4%29%5E2-7=0.


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%28z%2B4%29%5E2-7=0 Start with the given equation.


%28z%2B4%29%5E2=0%2B7Add 7 to both sides.


%28z%2B4%29%5E2=7 Combine like terms.


x%2B4=0%2B-sqrt%287%29 Take the square root of both sides.


z%2B4=sqrt%287%29 or z%2B4=-sqrt%287%29 Break up the "plus/minus" to form two equations.


z%2B4=sqrt%287%29 or z%2B4=-sqrt%287%29 Simplify the square root.


z=-4%2Bsqrt%287%29 or z=-4-sqrt%287%29 Subtract 4 from both sides.


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Answer:


So the solutions are z=-4%2Bsqrt%287%29 or z=-4-sqrt%287%29


which approximate to z=-1.354 or z=-6.646