Question

1. if $a + b = s$ and $a - b = t$, then which of the following expresses the value of $ab$ in terms of $s$ and $t$?

a. $st$
b. $\frac{(s-t)}{2}$
c. $\frac{(s+t)}{2}$
d. $\frac{(s^2 - t^2)}{4}$

Explanation:

Step1: Square both given equations

$$s^2=(a+b)^2=a^2+2ab+b^2$$
$$t^2=(a-b)^2=a^2-2ab+b^2$$

Step2: Subtract $t^2$ from $s^2$

$$s^2-t^2=(a^2+2ab+b^2)-(a^2-2ab+b^2)$$
$$s^2-t^2=4ab$$

Step3: Solve for $ab$

$$ab=\frac{s^2-t^2}{4}$$

Answer:

D. $\frac{(s^2 - t^2)}{4}$