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How do you factor the expression -12a^2+20a-21

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2 Answers

Hi Sam,
 
It would be impossible to completely factor -12a2+20a-21 into the product of real numbers. The expression would be considered prime in that case. This is probably the answer you are looking for.
 
However, factoring would be possible if complex linear factors are allowed. It would require using the definition i = √(-1) and the quadratic formula:
 
a = (-B ± √(B2-4AC))/2A
 
My first step would be to factor out -1/3, turning A into a positive square number:
 
-12a2+20a-21 = -1/3(36a2-60a+63)
 
From there, I would use the quadratic formula with A=36, B=-60 and C=63:
 
a = (-(-60) ± √((-60)2-4(36)(63)))/(2(36))
= (60 ± √(3600-9072))/72                   [Simplifying]
= (60 ± √(-5472))/72                          [Subtracting]
= (60 ± 12i√(38))/72                           [Simplifying the radical]
= 12(5 ± i√(38))/72                            [Factoring the numerator]   
= (5 ± i√(38))/6                                 [Reducing]
 
With a bit of rewriting this would give a factorization of:
 
-1/3(6a - 5 + i√(38))(6a - 5 - i√(38)) = -1/3(-6i a + √(38) + 5i)(6i a + √(38) - 5i)
 
Bonus:
 
It is also possible to complete the square using the expression -12a2+20a-21:
 
-12a2+20a-21 = -12(a2-5/3a)-21                               [Rewriting]
= -12(a2-5/3a + 25/36)-21 + 25/3     [Completing the square]
= -12(a-5/6)2-63/3+25/3                  [Rewriting]
= -12(a-5/6)2-38/3                           [Simplifying]
 
This form is important, primarily because it shows that at a = 5/6 the expression reaches a maximum value of -38/3.

This expression can not be factorized in the set of real numbers.
There is formula for factorization of quadratic expression: ax^2+bx+c=a(x-x,)(x-x,,) , where x, and x.. are roots of quadratic equation ax^2+bx+c=0 . To find the roots we can use the formula x = (-b ± √D)/2a , where D=b^2-4ac . There are 2 roots if D>0 , there is 1root if D=0 , and zero roots in set of real numbers if D is negative.
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Let's use information above: -12 a^2 + 20a - 21 = 0
D = 20^2 -4(-12)(-21) < 0