(a+b+c)(-a+b+c)+(2c+b-a)²-2(b+c)² simplify true Binomial

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Simplify `(a+b+c)(-a+b+c)+(2c+b-a)^2-2(b+c)^2` :

Lets look at each term separately:

We can use the distributive property on the first term:

`(a+b+c)(-a+b+c)`

`=-a^2+ab+ac-ab+b^2+bc-ac+bc+c^2` collecting like terms we get

`=-a^2+b^2+2bc+c^2`

To expand the second term rewrite as the product of trinomials:

`(2c+b-a)^2=(2c+b-a)(2c+b-a)`

`=4c^2+2bc-2ac+2bc+b^2-ab-2ac-ab+a^2`

`=4c^2+4bc-4ac+b^2-2ab+a^2`

We can expand the last term either by writing as a product of binomials or using `(x+y)^2=x^2+2xy+y^2` :

`-2(b+c)^2=-2[b^2+2bc+c^2]`

`=-2b^2-4bc-2c^2`

Putting the expanded terms together yields:

`(a+b+c)(-a+b+c)+(2c+b-a)^2-2(b+c)^2`

`=[-a^2+b^2+2bc+c^2]+[4c^2+4bc-4ac-2ab+a^2]+[-2b^2-4bc-2c^2]` `=2bc+3c^2-4ac-2ab` which is the answer.

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