Question 156·Hard·Equivalent Expressions

After you multiply both sides by $x^2 - 4$, you will get $x^4 - 10x^2 + k$ on the left and some product on the right. Write that product as $(x^2 - 6)(x^2 - 4)$.

Expand $(x^2 - 6)(x^2 - 4)$ carefully. Once you have it as a single polynomial, compare it to $x^4 - 10x^2 + k$ and focus on the constant term to find $k$.

In Desmos, type f(x) = (x^4 - 10x^2 + k)/(x^2 - 4) and g(x) = x^2 - 6. Replace k with a letter like a so you can use it as a slider: f(x) = (x^4 - 10x^2 + a)/(x^2 - 4).

Create a slider for a if Desmos does not add it automatically. Then manually set a equal to each of the answer choices (16, 20, 24, 28) one at a time, or type those specific values into the slider.

For the correct value of a, the graphs of f(x) and g(x) will lie exactly on top of each other for all $x$ except at $x = \pm 2$, where f(x) has holes because of the denominator. The value of a that makes the graphs coincide is the correct choice for $k$.

We are told

for all real $x$ such that $x^2 \ne 4$ (so $x^2 - 4 \ne 0$). Because the denominator is never zero in the allowed domain, we can safely multiply both sides by $x^2 - 4$:

Now we just need to expand the product on the right side.

Expand $(x^2 - 6)(x^2 - 4)$ step by step:

So the equation from Step 1 becomes

For two polynomials to be equal for all $x$, the coefficients of each matching power of $x$ must be the same. On the left we have $x^4 - 10x^2 + k$, and on the right $x^4 - 10x^2 + 24$.

The $x^4$ and $x^2$ coefficients already match. That means the constant terms must match as well:

So the value of $k$ is 24, which corresponds to choice C.