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Video: Expanding and Simplifying Algebraic Expressions

Tim Burnham

Expand and simplify (2𝑥 − 3)(3𝑥 − 4) − 2(4𝑥 − 5)(3𝑥 + 1).


Video Transcript

Expand and simplify two 𝑥 minus three times three 𝑥 minus four minus two times four 𝑥 minus five times three 𝑥 plus one.

Now in this question, we’ve got two sets of parentheses multiplied together here and two sets of parentheses multiplied together here, and then that result multiplied by two.

So the four 𝑥 is gonna be multiplied by the three 𝑥, but it’s also gonna be multiplied by the positive one. And the negative five is gonna be multiplied by the three 𝑥 and also by the positive one.

Now the whole result of this has still got to be multiplied by negative two, so I’m just still gonna write that out inside some big parentheses here. So now we’re doing four 𝑥 times three 𝑥 is 12𝑥 squared.

And four 𝑥 times positive one is just positive four 𝑥, negative five times three 𝑥 is negative 15𝑥, and negative five times positive one is negative five.

Now before I move on, I’ve got a little bit of tidying up to do. I’ve got negative eight 𝑥 take away another nine 𝑥, which is negative 17𝑥. Then I’ve got my positive 12 and we’ve got to take away two times the whole of this thing. And again in here, I’ve got four 𝑥 take away 15𝑥, which I can simplify. And four take away 15 is negative 11, so that’s negative 11𝑥.

And I’ve still got the negative five on the end there, so I’ve got six 𝑥 squared minus 17𝑥 plus 12 minus two times the whole of 12𝑥 squared minus 11𝑥 minus five.

So my next task is to multiply everything in the parentheses by negative two, so negative two times 12𝑥 squared, negative two times negative 11𝑥, and negative two times negative five.

Well negative two times positive 12𝑥 squared is negative 24𝑥 squared. And negative two times negative 11𝑥, negative times negative makes positive, so that’s positive 22𝑥. And negative two times negative five is positive 10.

So we’ve multiplied each element of that parenthesis at the end there by negative two, so now I’ve got six 𝑥 squared here and I’ve got to take away 24𝑥 squared. Well that’s negative 18𝑥 squared.

I’ve also got a negative 17𝑥 and I’m going to add 22𝑥, which gives me positive five 𝑥. And then finally, I’ve got positive 12 and I’m adding another 10, which gives me plus 22. So our answer is negative 18𝑥 squared plus five 𝑥 plus 22.

And we had to be really careful when we were doing that question. We had to remember when multiplying out the parentheses that we had to multiply each element of one by each element of the other.

And one of the hardest parts of this question was keeping track of all those negative signs and remembering that negative times negative makes a positive and negative times positive makes a negative, so writing your answer out in full is really vital to get the correct answer at the end of these questions.