SOLUTION: Sviatoslav solved the quadratic equation x^2-x-1=0 by completing the square. In the process, he came up with the equivalent equation (x+a)^2 = b, where a and b are constants. Wh

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Sviatoslav solved the quadratic equation x^2-x-1=0 by completing the square. In the process, he came up with the equivalent equation (x+a)^2 = b, where a and b are constants. Wh      Log On


   

Question 1086031: Sviatoslav solved the quadratic equation x^2-x-1=0 by completing the square. In the process, he came up with the equivalent equation (x+a)^2 = b, where a and b are constants.
What is b?
Found 2 solutions by josmiceli, MathLover1:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+x%5E2+-+x+-+1+=+0+
+%28+x+%2B+a+%29%5E2+=+b+
+x%5E2+%2B+2a%2Ax+%2B+a%5E2+-+b+=+0+
------------------------------
Comparing this with the original equation:
+2a+=+-1+
+a+=+-1%2F2+
and
+a%5E2+-+b+=+-1+
+%28-1%2F2%29%5E2+-+b+=+-1+
+1%2F4+-+b+=+-1+
+b+=+1+-+5%2F4+
+b+=+5%2F4+
----------------------
check:
+%28+x+-1%2F2+%29%5E2+=+5%2F4+
+x%5E2+-+x+%2B+1%2F4+=+5%2F4+
+x%5E2+-+x+-+1+=+0+

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2-x-1=0.....to complete square, recall %28a-b%29%5E2=a%5E2-2ab%2Bb%5E2;so, we need to add and subtract b%5E2
%28x%5E2-x%2Bb%5E2%29-b%5E2-1=0.. ..since coefficients a=1and 2ab=-1, we have 2%2A1%2Ab=-1->2b=-1->b=-1%2F2
%28x%5E2-x%2B%28-1%2F2%29%5E2%29-%28-1%2F2%29%5E2-1=0
%28x-1%2F2%29%5E2-%281%2F4%29-1=0
%28x-1%2F2%29%5E2-1%2F4-4%2F4=0
%28x-1%2F2%29%5E2-5%2F4=0
%28x-1%2F2%29%5E2=5%2F4