If 𝑥 minus 𝑎 multiplied by 𝑥 minus two equals 𝑥 squared minus six 𝑥 plus 𝑏 is valid for all values of 𝑥, what is the value of 𝑏?
Let’s firstly consider the two parentheses, or brackets, on the left-hand side of the equation, 𝑥 minus 𝑎 multiplied by 𝑥 minus two. We can expand, or distribute, these parentheses using the FOIL method. Multiplying the first terms, 𝑥 multiplied by 𝑥 gives us 𝑥 squared. Multiplying the outside terms gives us negative two 𝑥. Multiplying the inside terms gives us negative 𝑎𝑥. And finally, multiplying the last terms gives us two 𝑎.
Negative 𝑎 multiplied by negative two is equal to positive two 𝑎, as multiplying two negative terms gives a positive answer. We can then group, or collect, the two middle terms, negative two 𝑥 and negative 𝑎𝑥. Factorising out negative 𝑥 gives us negative of two plus 𝑎 multiplied by 𝑥. 𝑥 minus 𝑎 multiplied by 𝑥 minus two simplifies to 𝑥 squared minus two plus 𝑎 multiplied by 𝑥 plus two 𝑎. We now need to compare this to the right-hand side of the initial equation, 𝑥 squared minus six 𝑥 plus 𝑏.
We are told that the two sides of the equation are equal for all values of 𝑥. This means that the coefficients on the left must be equal to the coefficients on the right. The coefficient of 𝑥 on the left-hand side is negative two plus 𝑎. The coefficient of 𝑥 on the right-hand side is equal to negative six. Therefore, negative two plus 𝑎 is equal to negative six. We can multiply both sides of this equation by negative one. This gives us two plus 𝑎 is equal to six. Subtracting two from both sides of this equation gives us 𝑎 is equal to four. This means that the constant 𝑎 equals four.
If we now consider the constants, the left-hand side of the equation had a constant two 𝑎, and the right-hand side had a constant 𝑏. This means that two 𝑎 is equal to 𝑏. We have already worked out that 𝑎 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 𝑥 minus 𝑎 multiplied by 𝑥 minus two is equal to 𝑥 squared minus six 𝑥 plus 𝑏, then 𝑏 is equal to eight.
We could check this answer by substituting in 𝑎 equals four and then expanding the parentheses, 𝑥 minus four multiplied by 𝑥 minus two. Using the FOIL method once again, this gives us 𝑥 squared minus two 𝑥 minus four 𝑥 plus eight. We can then simplify this to 𝑥 squared minus six 𝑥 plus eight. Therefore, our answer for 𝑏 is correct.