Question

If a, b, c belong to R and equations $a{x}^2+bx+c=0and2x^2+4x+6=0$ have a common root. Also given that a + b + c = 18. Find the value of $a^{2}bc$

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Answer 1

Let $a{x}^2+bx+c=0$ ........................(1)

$2x^2+4x+6=0$ ..........................(2)

Since $D=b^2-4ac=16-4\times 2\times 6=-32-ve$

Roots should be imaginary

As the imaginary roots occurs in conjugate pairs. So, both the roots of the equation will be common.

So, $\frac{a}{2}=\frac{b}{4}=\frac{c}{6}$

a : b : c = 1 : 2 : 3

let a = x

b = 2x

c = 3x

a + b + c = 18

x + 2x + 3x = 18

6x = 18

x = 3

a = 3

b = 6

c = 9

The value of $a^{2}bc=3^2\times 6\times 9$

$=9\times 6\times 9$

= 486