Video: Algebraic Manipulation of a Quadratic Expression

If the expression (βˆ’5π‘₯Β² + 2π‘₯ βˆ’ 3) βˆ’ 3 (βˆ’3π‘₯Β² βˆ’ π‘₯ βˆ’ 2) is rewritten in the form π‘Žπ‘₯Β² + 𝑏π‘₯ + 𝑐, where π‘Ž, 𝑏, and 𝑐 are constants, what is the value of π‘Ž?

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Video Transcript

If the expression negative five π‘₯ squared plus two π‘₯ minus three minus three times negative three π‘₯ squared minus π‘₯ minus two is rewritten in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐, where π‘Ž, 𝑏, and 𝑐 are constants, what is the value of π‘Ž?

The first thing we’ll do is carefully copy down the expression we’re working with. Because we can’t simplify what’s in the first set of parentheses any further, we can bring it down outside of the parentheses and just say negative five π‘₯ squared plus two π‘₯ minus three. But the second set of parentheses is being multiplied by negative three. And that means we need to distribute this negative three across all terms inside the second set of parentheses.

To do that, we multiply negative three by negative three π‘₯ squared, and we get positive nine π‘₯ squared. And then, we multiply negative three by negative π‘₯ to get positive three π‘₯. And then, negative three times negative two equals positive six. Remember, our goal is to get something in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐. This means we can only have one term that is in the form of π‘Žπ‘₯ squared. Currently, we have two π‘₯ squared terms, negative five π‘₯ squared and nine π‘₯ squared.

We call these like terms and we can combine them. We combine these like terms by adding the coefficients together. Negative five plus nine equals four. And that means negative five π‘₯ squared plus nine π‘₯ squared equals four π‘₯ squared. This question has only asked us for the value of π‘Ž. π‘Ž is the constant coefficient for the π‘₯ squared term. The constant coefficient for the π‘₯ squared term of this expression is four.

Without going any further, we can say that π‘Ž equals four as we weren’t required to finish evaluating the expression. If we did evaluate the rest of the expression, we would have two π‘₯ plus three π‘₯, which equals five π‘₯. And negative three plus six equals positive three. The expression rewritten in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 would look like this, four π‘₯ squared plus five π‘₯ plus three. And again we can see that in the π‘Ž position, the π‘Ž value is four.

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