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If $23$ and ‒3 are the roots of the quadratic equation $a x 2 + 7 x + b$ = 0, then find the values of a and b.

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3, -6

4, 6

1, 2

3, -5

answer is A.

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Detailed Solution

We are given the quadratic polynomial equation as

a x 2 + 7 x + b = 0

and

23

and ‒3 are the roots. We have to determine the values of a and b.
The relation between the zeros, say

α

and

β

, and the coefficients of a quadratic polynomial

a x 2 + b x + c

is given as follows,

α + β = − b a α β = c a

So, comparing the equation

a x 2 + 7 x + b = 0

with the general form, we get the coefficients as a = a, b = 7 and c = b. Now, as per the relation between the zeros and the coefficients of a quadratic polynomial, we have

α + β = − 7 a α β = b a

Next, we have the roots of the given equation.
So,

α = 23

and

β = − 3

.
Hence, we can rewrite the above relations as follows,

23 + (− 3) = − 7 a .......... (i) 23 × (− 3) = b a .......... (i i)

Now solving equation (i), we get,

23 + (− 3) = − 7 a 2 − 9 3 = − 7 a − 7 3 = − 7 a a = 3

Now, substituting this value of a in the equation (ii), we get,

23 × (− 3) = b 3 − 2 = b 3 − 2 × 3 = b

b = − 6

So, the values of a and b are obtained as 3 and –6 respectively.
Hence, the correct option is (1).

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