Question

Equating coefficients to find constants in an identity requiring bracket expansion

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Answer 1

Solving the equation yields
`\(q = (84x - 18)/(21x)\)` in terms of x, and choosing
x = 0 gives p = 3. Therefore, p = 3 and
`\(q = (84x - 18)/(21x)\)`.

Let's expand the parentheses in the equation:

3(3qx + p) + 3(4qx + p) = 84x + 18

9qx + 3p + 12qx + 3p = 84x + 18

Combine like terms:

21qx + 6p = 84x + 18

Move the x term to the other side of the equation:

21qx = 84x - 6p

Isolate q:

q = (84x - 6p) / 21x

This is the equation for q in terms of x and p. To solve for p, we need another equation.

We can use the fact that the given equation is true for any value of x. Let's try to choose a value of x that will help us eliminate q.

For example, if we choose x = 0, the equation becomes:

6p = 18

p = 3

Now we can substitute this value of p into the equation for q:

q = (84x - 6(3)) / 21x

q = (84x - 18) / 21x

This is the equation for q in terms of x only.

Therefore, the values of p and q are:

p = 3

q = (84x - 18) / 21x