### Video Transcript

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

Letβs firstly consider the two parentheses, or brackets, on the left-hand side of the equation, π₯ minus π 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 ππ₯. And finally, multiplying the last terms gives us two π.

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

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. The coefficient of π₯ on the right-hand side is equal to negative six. Therefore, negative two plus π is equal to negative six. We can multiply both sides of this equation by negative one. This gives us two plus π is equal to six. Subtracting two from both sides of this equation gives us π is equal to four. This means that the constant π equals four.

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

We could check this answer by substituting in π 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 π is correct.