Question

If a + b = 7 and ab = 12 then what is the value of $a^2 - ab - b^2$?

• A. 12
• B. 13
• C. 14
• D. -5

Answer 1

Answer: (D)

-5

Answer 2

Step 1: Let $S = a + b = 7$ and $P = ab = 12$.
Step 2: The quadratic equation with roots $a$ and $b$ is

$$t^2 - S\,t + P = t^2 - 7t + 12 = 0,$$

whose roots are $t = 3$ and $t = 4$.
Step 3: Assuming $a = 4$ and $b = 3$, compute

$$a^2 - ab - b^2 = 4^2 - (4)(3) - 3^2 = 16 - 12 - 9 = -5.$$

Hence, the required value is $-5$.