Video: US-SAT04S4-Q35-525104126427

If (π‘₯ βˆ’ π‘Ž)(π‘₯ βˆ’ 2) = π‘₯Β² βˆ’ 6π‘₯ + 𝑏 is valid for all values of π‘₯, what is the value of 𝑏?

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Video Transcript

If π‘₯ minus π‘Ž multiplied by π‘₯ minus two equals π‘₯ squared minus six π‘₯ plus 𝑏 is valid for all values of π‘₯, what is the value of 𝑏?

Let’s firstly consider the two parentheses, or brackets, on the left-hand side of the equation, π‘₯ minus π‘Ž multiplied by π‘₯ minus two. We can expand, or distribute, these parentheses using the FOIL method. Multiplying the first terms, π‘₯ multiplied by π‘₯ gives us π‘₯ squared. Multiplying the outside terms gives us negative two π‘₯. Multiplying the inside terms gives us negative π‘Žπ‘₯. And finally, multiplying the last terms gives us two π‘Ž.

Negative π‘Ž multiplied by negative two is equal to positive two π‘Ž, as multiplying two negative terms gives a positive answer. We can then group, or collect, the two middle terms, negative two π‘₯ and negative π‘Žπ‘₯. Factorising out negative π‘₯ gives us negative of two plus π‘Ž multiplied by π‘₯. π‘₯ minus π‘Ž multiplied by π‘₯ minus two simplifies to π‘₯ squared minus two plus π‘Ž multiplied by π‘₯ plus two π‘Ž. We now need to compare this to the right-hand side of the initial equation, π‘₯ squared minus six π‘₯ plus 𝑏.

We are told that the two sides of the equation are equal for all values of π‘₯. This means that the coefficients on the left must be equal to the coefficients on the right. The coefficient of π‘₯ on the left-hand side is negative two plus π‘Ž. The coefficient of π‘₯ on the right-hand side is equal to negative six. Therefore, negative two plus π‘Ž is equal to negative six. We can multiply both sides of this equation by negative one. This gives us two plus π‘Ž is equal to six. Subtracting two from both sides of this equation gives us π‘Ž is equal to four. This means that the constant π‘Ž equals four.

If we now consider the constants, the left-hand side of the equation had a constant two π‘Ž, and the right-hand side had a constant 𝑏. This means that two π‘Ž is equal to 𝑏. We have already worked out that π‘Ž is equal to four. Therefore, two multiplied by four is equal to 𝑏. Two times four is equal to eight. Therefore, the value for 𝑏 is eight. If π‘₯ minus π‘Ž multiplied by π‘₯ minus two is equal to π‘₯ squared minus six π‘₯ plus 𝑏, then 𝑏 is equal to eight.

We could check this answer by substituting in π‘Ž equals four and then expanding the parentheses, π‘₯ minus four multiplied by π‘₯ minus two. Using the FOIL method once again, this gives us π‘₯ squared minus two π‘₯ minus four π‘₯ plus eight. We can then simplify this to π‘₯ squared minus six π‘₯ plus eight. Therefore, our answer for 𝑏 is correct.

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