Product Formulas — Hard Practice Quiz

A Algebra cheat sheet for Product Formulas — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. Given that $a+b=P$ and $a^2+b^2=Q$, express $ab$ in terms of $P$ and $Q$.

   • $\frac{P^2 - Q}{2}$
   • $\frac{Q - P^2}{2}$
   • $P^2 - Q$
   • $Q - P^2$

   Answer: $\frac{P^2 - Q}{2}$

2. Simplify the expression $(x+y)^2 - (x-y)^2$.

   • $2x^2+2y^2$
   • $4xy$
   • $2xy$
   • $0$

   Answer: $4xy$

3. If $a+b+c=0$, what is the value of $a^2+b^2+c^2$ in terms of $ab, bc, ca$?

   • $2(ab+bc+ca)$
   • $-2(ab+bc+ca)$
   • $ab+bc+ca$
   • $0$

   Answer: $-2(ab+bc+ca)$

4. Given $x - \frac{1}{x} = 3$, find the value of $x^3 - \frac{1}{x^3}$.

   • $27$
   • $30$
   • $36$
   • $42$

   Answer: $36$

5. If $a+b=4$ and $a^2+b^2=10$, what is the value of $a^3+b^3$?

   • $28$
   • $32$
   • $40$
   • $64$

   Answer: $28$

6. Expand and simplify $(x+y+1)^2 - (x+y)^2$.

   • $2x+2y$
   • $2x+2y+1$
   • $1$
   • $x^2+y^2+1$

   Answer: $2x+2y+1$

7. If $a^2+b^2=2ab$, what is the value of $(a-b)^3$?

   • $0$
   • $1$
   • $a^3-b^3$
   • $a^3+b^3$

   Answer: $0$

8. Given $x+y=A$ and $x-y=B$, express $x^2+y^2$ in terms of $A$ and $B$.

   • $\frac{A^2+B^2}{2}$
   • $\frac{A^2-B^2}{2}$
   • $A^2+B^2$
   • $A^2-B^2$

   Answer: $\frac{A^2+B^2}{2}$

9. If $x+y=5$ and $x^2+y^2=13$, find the value of $(x-y)^2$.

   • $1$
   • $25$
   • $13$
   • $12$

   Answer: $1$

10. Simplify $(a+b)^3 - (a^3+b^3)$.

    • $3ab(a+b)$
    • $3ab$
    • $0$
    • $a^3+b^3$

    Answer: $3ab(a+b)$