# SAT Math Multiple Choice Question 941: Answer and Explanation

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**Question: 941**

**9.** If *x* – 4 is a factor of *x*^{2} – *kx* + 2*k*, where *k* is a constant, what is the value of *k*?

- A. –4
- B. 4
- C. 8
- D. 12

**Correct Answer:** C

**Explanation:**

**C**

**Difficulty:** Hard

**Category:** Passport to Advanced Math / Quadratics

**Strategic Advice:** You could substitute each of the answer choices for *k* and factor the resulting expression, but this will use up valuable time on Test Day. Instead, think about what it means for *x* - 4 to be a *factor* of the expression.

**Getting to the Answer:** If *x* - 4 is a factor of *x*^{2} - *kx* + 2*k*, then *x*^{2} - *kx* + 2*k* can be written as the product (*x* - 4)(*x* - *a*) for some real number *a*. Expanding the product (*x* - 4)(*x* - *a*) yields *x*^{2} - 4*x* - *ax* + 4*a*, which can be rewritten as *x*^{2} - (4 + *a*)*x* + 4*a*. Substituting this for the factored form of the original expression results in the equation *x*^{2} - (4 + *a*)*x* + 4*a* = *x*^{2} - *kx* + 2*k*. Two quadratic equations are equal if and only if the coefficients of their like terms are equal, so 4 + *a* = *k* and 4*a* = 2*k*. You now have a system of equations to solve. The first equation is already solved for *k*, so substitute 4 + *a* into the second equation for *k* and solve for *a*:

The questions asks for the value of *k*, and *k* = 4 + *a*, so *k* = 4 + 4 = 8, which is (C).