Question

solve the above question.​

Answer 1

Given:-

• ${{x}^{2} - (2b - 1)x + ( {b}^{2} - b - 20) = 0}$

To find:-

• The value of x in terms of b.

Answer:-

Given that,

${{x}^{2} - (2b - 1)x + ( {b}^{2} - b - 20) = 0}$

Splitting the middle term of (b² - b - 20) term,

$\longrightarrow {x}^{2} - (2b - 1)x + \Big \{ {b}^{2} - 5b + 4b - 20 \Big \} = 0$

$\longrightarrow {x}^{2} - (2b - 1)x + \Big \{ b(b - 5) + 4(b - 5) \Big \} = 0$

$\longrightarrow {x}^{2} - (2b - 1)x + (b - 5)(b + 4) = 0$

We observe that, (b - 5) + (b + 4)= 2b - 1. Hence, we can split the middle term that is (2b - 1)x as follows,

$\longrightarrow {x}^{2} - \Big \{(b - 5) + (b + 4) \Big \}x + (b - 5)(b + 4) = 0$

$\longrightarrow {x}^{2} - (b - 5)x - (b + 4) x + (b - 5)(b + 4) = 0$

$\longrightarrow x \Big \{x - (b - 5) \Big \} - (b + 4) \Big \{ x - (b - 5) \Big \} = 0$

$\longrightarrow \Big \{x - (b - 5) \Big \} \Big \{ x - (b + 4) \Big \} = 0$

So

either {x - (b - 5)} = 0,

or {x - (b + 4)} = 0

If {x - (b - 5)} = 0:

$x - (b - 5) = 0$

$\longrightarrow x = b - 5$

If {x - (b + 4)} = 0:

$x = b + 4$

$\longrightarrow x = b + 4$

Hence,

$\longrightarrow \underline{ \underline{x = b - 5 \: or \: x = b + 4}}$