Video: Factorizing a Difference of Two Squares

Factorize fully (π‘₯ + 4𝑦 + 3)Β² βˆ’ (π‘₯ βˆ’ 4𝑦 βˆ’ 3)Β².

02:41

Video Transcript

Factorize fully π‘₯ plus four 𝑦 plus three squared minus π‘₯ minus four 𝑦 minus three squared.

The key here is to notice that one square term is being subtracted from another square term. And we call this the difference of squares. π‘Ž squared minus 𝑏 squared is equal to π‘Ž plus 𝑏 times π‘Ž minus 𝑏. We can use this to help us factorize.

For us, π‘Ž equals π‘₯ plus four 𝑦 plus three. And 𝑏 equals π‘₯ minus four 𝑦 minus three. We need π‘Ž plus 𝑏 times π‘Ž minus 𝑏, which would look like this. The key here is to be really careful with our positive and negative signs, π‘₯ plus four 𝑦 plus three plus π‘₯ minus four 𝑦 minus three and then π‘₯ plus four 𝑦 plus three. This time, we’re subtracting. So we need to distribute the subtraction. Minus π‘₯ minus negative four 𝑦 equals plus four 𝑦. And minus negative three equals plus three.

The next step will be to combine like terms. π‘₯ plus π‘₯ equals two π‘₯. Positive four 𝑦 minus four 𝑦 cancels out. Positive three minus three also cancels out. And that means our π‘Ž plus 𝑏 term has been simplified to two π‘₯.

On the other side, we have the π‘Ž minus 𝑏 terms. Positive π‘₯ minus π‘₯ cancels out. Four 𝑦 plus four 𝑦 equals eight 𝑦. And three plus three is six. Our π‘Ž minus 𝑏 terms equals eight 𝑦 plus six. But eight 𝑦 and six share a common factor. They both have a factor of two. If we take out that common factor of two, we could rewrite this to say, two times four 𝑦 plus three.

And remember, we’re multiplying the π‘Ž plus 𝑏 times the π‘Ž minus 𝑏. We would then have two π‘₯ times two times four 𝑦 plus three. We can multiply this, two times two, which equals four. Bring down the π‘₯. And then bring down the four 𝑦 plus three.

The fully factorized form is four π‘₯ times four 𝑦 plus three.

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