Question

Form the equations whose roots are
(1) $-\frac{4}{5},\frac{3}{7}$
(2) $\frac{m}{n},-\frac{n}{m}$
(3) $\frac{p-q}{p+q},\frac{p+q}{p-q}$
(4) $7\pm 2\sqrt{5}$
(5) $\pm 2\sqrt{3}-5$
(6) $-p\pm 2\sqrt{2q}$
(7) $-3\pm 5i$
(8) $-a\pm ib$
(9) $\pm i(a-b)$
(10)$-3,\frac{2}{3},\frac{1}{2}$
(11)$\frac{a}{2},0,-\frac{2}{a}$

Answer 1

1. Roots are $-\frac{4}{5},\frac{3}{7}$

Sum of roots is $-\frac{13}{35}$ and product of roots is $-\frac{12}{35}$

Therefore the equation is $35x^2+13x-12=0$

2. Roots are $\frac{m}{n},-\frac{n}{m}$

Sum of roots is $\frac{m^2-n^2}{mn}$ and product of roots is $-1$

Therefore the equation is $mn{x}^2+(n^2-m^2)x-mn=0$

3. Roots are $\frac{p-q}{p+q},\frac{p+q}{p-q}$

Sum of roots is $\frac{2(p^2+q^2)}{p^2-q^2}$ and product of roots is $1$

Therefore the equation is $(p^2-q^2)x^2-2(p^2+q^2)x+p^2-q^2=0$

4. Roots are $7+2\sqrt{5},7-2\sqrt{5}$

Sum of roots is $14$ and product of roots is $29$

Therefore the equation is $x^2-14x+29=0$

5. Roots are $-5+2\sqrt{3},-5-2\sqrt{3}$

Sum of roots is $-10$ and product of roots is $13$

Therefore the equation is $x^2+10x+13=0$

6. Roots are $-p+2\sqrt{2q},-p-2\sqrt{2q}$

Sum of roots is $-2p$ and product of roots is $p^2-8q$

Therefore the equation is $x^2+2px+p^2-8q=0$

7. Roots are $-3+5i,-3-5i$

Sum of roots is $-6$ and product of roots is $34$

Therefore the equation is $x^2+6x+34=0$

8. Roots are $-a+ib,-a-ib$

Sum of roots is $-2a$ and product of roots is $a^2+b^2$

Therefore the equation is $x^2+2ax+a^2+b^2=0$

9. Roots are $-i(a-b),i(a-b)$

Sum of roots is $0$ and product of roots is ${(a-b)}^2$

Therefore the equation is $x^2+{(a-b)}^2=0$

10. Roots are $-3,\frac{2}{3},\frac{1}{2}$

Sum of roots is $-\frac{11}{6}$, sum of product of roots taken two at a time is $-\frac{19}{6}$ and product of roots is $-1$

Therefore the equation is $6x^3+11x^2-19x+6=0$

11. Roots are $\frac{a}{2},0,-\frac{2}{a}$

Sum of roots is $\frac{a^2-4}{2a}$, sum of product of roots taken two at a time is $-1$ and product of roots is $0$

Therefore the equation is $2a{x}^3+(4-a^2)x^2-2ax=0$