Here is the procedure for factoring any quadratic trinomial using the completing the square method:
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x 2 + 6 x + 9 − 9 − 5.
(x 2 + 6 x + 9) − 9 − 5.
(x + 3) 2 − 14.
a 2 − b 2 = (a + b) (a − b).
The right side of this formula gives the expression in factored form.
2 ( x 2 + 4.5 x − 10)
2 (x 2 + 4.5 x + 5.0626 − 5.0625 − 10)
2 ( [x 2 + 4.5 x + 5.0626] − 5.0625 − 10)
2 ( [x + 2.25] 2 − 15.0625)
2 ( [x + 2.25] 2 − 3.88 2 )
a 2 − b 2 = (a + b) (a − b).
[x + 2.25] 2 − 3.88 2 = (x + 2.25 + 3.88) (x + 2.25 − 3.88)
2 (x + 6.13) (x − 1.63)
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The quantity b 2 − 4 a c inside the square root is called the discriminant, and is denoted with the letter D. If D is negative then the above formula contains square roots of negative numbers. Over the real numbers this is not allowed and means that the quadratic trinomial can't be factored. But if we are doing algebra over the complex numbers then the square root of a negative number is simply an imaginary number. Thus the only complication is that the factors contain imaginary numbers.
x 2 + 4 x + 4 − 4 + 14.
(x 2 + 4 x + 4) − 4 + 14.
(x + 2) 2 + 10.
a 2 + b 2 = (a + b i ) (a − b i ).
(x + 2) 2 − (−10)
a 2 − b 2 = (a + b) (a − b).
The right side of this formula gives the expression in factored form.
2 ( x 2 + 4.5 x + 7 )
2 (x 2 + 4.5 x + 5.0626 − 5.0625 + 7 )
2 ( [x 2 + 4.5 x + 5.0626] − 5.0625 + 7 )
2 ( [x + 2.25] 2 + 1.9375)
2 ( [x + 2.25] 2 + 1.39 2 )
a 2 + b 2 = (a + b i ) (a − b i ).
[x + 2.25] 2 + 1.39 2 = (x + 2.25 + 1.39 i ) (x + 2.25 − 1.39 i )
2 (x + 2.25 + 1.39 i ) (x + 2.25 − 1.39 i )
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