Question

If $x=\frac{2}{3}$ and x = - 3 are the roots of equation $a{x}^2+7x+b=0,$ find values of a and b.

Answer 1

for a polynomial $a_1{x}^2+b_{1}x+c_1=0$ sum of the roots is $\frac{-b_1}{a_1}$
and the products of the roots is $\frac{c_1}{a_1}$

hence here sum of the roots is $\frac{-7}{a}=$ $x_1+x_2$ = $-3+\frac{2}{3}=\frac{-7}{3}$


hence $\frac{-7}{a}$ = $\frac{-7}{3}$ hence $a=3$

and product of the roots is $x_1\times x_2=\frac{2}{3}\times (-3)=-2=\frac{b}{3}$(from the given equation)

hence $b=-6$


so $a=3$ and $b=-6$