Write the given equation in the form $ax^2+bx+c=0$ and identify $a$ (the coefficient of $x^2$) and $c$ (the constant term). Which part of $81x^2 + (81q - p)x - pq$ is $c$?

Once you have the product of the solutions written as a fraction involving $pq$ and 81, compare that product to the form $-k pq$. What value of $k$ makes the two expressions exactly the same?

In Desmos, assign simple positive values to the parameters, for example type p = 2 and q = 5. This turns the equation into a specific quadratic with numbers.

Enter y = 81x^2 + (81q - p)x - pq in Desmos. With your chosen $p$ and $q$, Desmos will graph a parabola.

Click on each x-intercept to see the two solution values. Then, in a new expression line, type their product (for example, A = (x_1)*(x_2) using the decimal values shown) so Desmos calculates the product of the roots as a number.

In separate lines, compute $-k \cdot p \cdot q$ for each answer choice by substituting that choice’s $k$ value into an expression like B = -(k)*p*q.

Compare each of these results to the product of the roots you found in Step 2; the answer choice whose $-k pq$ value matches the product is the correct $k$.

Recall that for any quadratic equation $ax^2+bx+c=0$ with solutions (roots) $r_1$ and $r_2$, the product of the solutions is given by

We will apply this relationship to the given equation.

Compare the given equation

to the standard form $ax^2+bx+c=0$.

• The coefficient of $x^2$ is $a = 81$.
• The constant term is $c = -pq$.

So, using the product-of-roots formula, the product of the two solutions is

From Step 2, the product of the solutions is $-\dfrac{pq}{81}$.

The problem states that this product can be written as $-k pq$, so set the two expressions equal:

Divide both sides by $-pq$ (which is positive since $p$ and $q$ are positive constants):

Therefore, the value of $k$ is $\dfrac{1}{81}$.