Question

1. (292) 10--(1204)b find b​

Answer 1

The value of b = 6

Given :

$\displaystyle \sf{ (292) _{10} = (1204)_{b}}$

To find :

The value of b

Solution :

Step 1 of 3 :

Write down the given equation

Here the given equation is

$\displaystyle \sf{ (292) _{10} = (1204)_{b}}$

Step 2 of 3 :

Form the equation to find the value of b

$\displaystyle \sf{ (292) _{10} = (1204)_{b}}$

Which gives

$\displaystyle \sf{292 =( 1 \times {b}^{3} ) +( 2 \times {b}^{2} ) +( 0 \times {b}^{1} ) +( 4 \times {b}^{0} ) }$

$\displaystyle \sf{ \implies {b}^{3} + 2 {b}^{2} + 4 = 292 }$

Step 3 of 3 :

Find the value of b

$\displaystyle \sf{ {b}^{3} + 2 {b}^{2} + 4 = 292 }$

$\displaystyle \sf{ \implies {b}^{3} + 2 {b}^{2} - 288 = 0 }$

$\displaystyle \sf{ \implies {b}^{3} - 6 {b}^{2} + 8 {b}^{2} - 48b + 48b - 288 = 0 }$

$\displaystyle \sf{ \implies {b}^{2} (b - 6) + 8b(b - 6) + 48(b - 6) = 0}$

$\displaystyle \sf{ \implies (b - 6)( {b}^{2} + 8b + 48) = 0}$

Now ,

b - 6 = 0 gives b = 6

Now for the equation b² + 8b + 48 = 0

Discriminant

= 8² - (4 × 1 × 48)

= 64 - 192

= - 128 < 0

So both roots of b² + 8b + 48 = 0 are imaginary

Therefore only possible value of b is 6

Hence the required value of b = 6

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