Question

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

Answer 1

Que $x=\frac{2}{3}\&x=-3.$ are roots of equation

$a{x}^2+7x+b=0,$ find value a & b

$\Rightarrow f\left(x\right)=a{x}^2+7x+b=0$

$\Rightarrow f\left(2/3\right)=a(\frac{2}{3}{)}^2+7\times \frac{2}{3}+b=0$

$\therefore a\times \frac{4}{9}+\frac{14}{3}+b=0$

$\therefore \frac{4a+42+9b}{9}=0$

$\therefore 4a+9b+42=\mathrm{0...}\left(i\right)$

$\Rightarrow f(-3)=a(-3{)}^2+7(-3)+b=0$

$9a-21+b=0...\left(ii\right)$

solving equation (i) & (ii)

$\therefore 4a+9b+42=0$

$\therefore 81a-189+9b=0$

$(4a-819)+(42+189)+(9b-4b)=0$

$\therefore -77a+231=0$

$\therefore 77a=231$

$\therefore a=\frac{231}{77}=3$

$\to$ substitute value of a in equation(ii)

$\therefore 9\times 3-21+b=0$

$\therefore b=21-27$

$\therefore b=-6$