Question

If x=2/3 and x=-3 are the roots of the quadratic equation (ax2 +7x +b=0).then find the values of a and b.

Answer 1

Answer: The value of a and b is 3 and -6 respectively.

Step-by-step explanation:

Since we have given that

$x=\frac{2}{3}\ and\ x=-3$ are the roots of the qadratic equation :

$ax^2+7x+b=0$

First we put the value of $x=\frac{2}{3}$ in the above quadratic equation:

$a(\frac{2}{3})^2+7\times \frac{2}{3}+b=0\\\\\frac{4a}{9}+\frac{14}{3}+b=0\\\\\frac{4a}{9}+b=\frac{-14}{3}\\\\4a+9b=-14\times 3=-42\\\\4a+9b=-42-----------(1)$

similarly, if we put the value of x = -3 in the quadratic equation:

$-3^2a+7\times -3+b=0\\\\9a-21+b=0\\\\9a+b=21\\\\b=21-9a-------------(2)$

From Eq(1) and (2), we get that

$4a+9b=-42\\\\4a+9(21-9a)=-42\\\\4a+189-81a=-42\\\\-77a=-42-189\\\\-77a=-231\\\\a=\frac{231}{77}=3$

Now, we put the value of 'a' in Eq(2), we have

$b=21-9a\\\\b=21-9\times 3\\\\b=21-27\\\\b=-6$

Hence, the value of a and b is 3 and -6 respectively.

Answer 2

Answer:

a = 3 ,  b = (-6)

Step-by-step explanation:

The given equation is ax^2 + 7x + b = 0

Its roots are given as x1 = -3 , x2 =  2/3

Now,

x1 + x2 = (-b / a)

-3 + 2 / 3 = ( -7 / a)

-9 + 2 / 3 = (-7 / a)

-7 / 3 = -7 / a

a = 3

Also,

x1 * x2 = b / a

-3 * 2 / 3 = b / 3

-2 = b / 3

b= (-6)

Thus the values of a and b are 3 , -6 respectively

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