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I have never learned this in school, I only learned algebra when you have $x$ and numbers, in equations, like this:

$$2x = 5(-2 + 5x)^2$$

I can solve that, but I cannot solve this one:

$$-3(7 -2x)^2 -5(1 + x)^2$$

How can I solve this one?

On the book, answer is: $$-17x^2 +74x -152$$

amWhy
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Roy
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  • the strategy to simplify both expressions (although the first one is an equation, while the second is not) is the same. – Avitus Nov 11 '13 at 15:01

3 Answers3

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Your expression, unlike the first equation, is not an equation.

It can be simplified by expanding the expression:

$$-3(7 -2x)^2 -5(1 + x)^2 = -3(49 - 28x + 4x^2) - 5(1 + 2x + x^2) = \cdots$$

Can you take it from here? Just multiply each expression in parentheses by the factors $-3$, $-5$, respectively, $$-3(49+4x^2-28x) - 5(1+x^2+2x) = -147 -12x^2 + 84x - 5 - 5x^2 - 10x$$ and now combine like terms. You'll arrive at $$-17x^2 + 74x - 152$$ which cannot be further simplified.

amWhy
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Let us do it. We have

$$-3(7 -2x)^2 -5(1 + x)^2=-3(7 -2x)(7-2x) -5(1 + x)(1+x)= -3(49-28x+4x^2)-5(1+x^2+2x)=-147+84x-12x^2-5-5x^2-10x=-152+74x-17x^2.$$

Avitus
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It looks like the problem is just to simplify the given polynomial $$-3(7 -2x)^2 -5(1 + x)^2.$$ This can be done using the identity $(a+b)^2 = a^2+b^2+2ab$. What you get is $$ -3(49+4x^2-28x) - 5(1+x^2+2x) = -147 -12x^2 + 84x - 5 - 5x^2 - 10x $$

which is in turn equal to the result you posted.

PS: This is not linear algebra! The squared terms are not linear.

Abramo
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