Question

If x=2/3 and x=-3 are the roots of the quadratic equation ax^2+7x+b=0 then find the value of a and b.

Answer 1

Answer

The value of a is 3 and that of b is -6

$\bf \large \underline{Given : }$

• The roots of the quadratic equation ax² + 7x + b = 0 are 2/3 and -3

$\bf \large \underline{To \: Find : }$

• The value of a and b

$\bf \large \underline{Solution : }$

We are given the quadratic equation :

ax² + 7x + b = 0

and 2/3 and -3 are its roots

From the relation of sum of roots and coefficients we have ,

$\sf sum \: of \: the \: roots = - \dfrac{coefficient \: of \: x {} }{coefficient \: of \: {x}^{2} }$

$\sf\implies \dfrac{2}{3} + (-3) = -\dfrac{7}{a} \\\\ \sf\implies \dfrac{2-9}{3} = -\dfrac{7}{a} \\\\ \sf\implies -\dfrac{7}{3}=-\dfrac{7}{a} \\\\ \sf\implies \dfrac{1}{a}=\dfrac{1}{3} \\\\ \sf\implies a = 3$

Again from relation of product of roots and coefficients we have ,

$\sf product \: of \: roots = \dfrac{constant \: term}{coefficient \: of \: {x}^{2} }$

$\sf\implies \dfrac{2}{3}\times (-3) = \dfrac{b}{a} \\\\ \sf\implies -2 = \dfrac{b}{3} \: \: \: \: \{ \because a = 3 \} \\\\ \sf\implies b = -6$

Therefore , the values are :

a = 3 and b = -6