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Question
Solve:
`[12x^2 + 18x + 42]/[ 18 x^2 + 12x +58 ] = [ 2x + 3]/[ 3x + 2]`
Solution
`[12x^2 + 18x + 42]/[ 18 x^2 + 12x +58 ] = [ 2x + 3]/[ 3x + 2]`
If x = 0, then `[12 xx 0 + 18 xx 0 + 42]/[ 18 xx 0 + 12 xx 0 +58 ] = [ 2 xx 0+ 3]/[ 3 xx 0 + 2] ⇒ 42/58 = 3/2`, which is not true.
So, x = 0 is not a solution of the given equation.
Now,
`[12x^2 + 18x + 42]/[ 18 x^2 + 12x +58 ] = [ 2x + 3]/[ 3x + 2] = [6x ( 2x + 3)]/[6x( 3x + 2)]` ....(by Theorem of equal ratios)
⇒ `[12x^2 + 18x + 42]/[ 18 x^2 + 12x +58 ] = [ 2x + 3]/[ 3x + 2] = [(12x^2 + 18x + 42) - 6x ( 2x + 3)]/[(18 x^2 + 12x +58) - 6x( 3x + 2)]`
⇒ `[12x^2 + 18x + 42]/[ 18 x^2 + 12x +58 ] = [ 2x + 3]/[ 3x + 2] = (12x^2 + 18x + 42 - 12x^2 + 18x )/(18 x^2 + 12x +58 - 18x^2 - 12x )`
⇒ `[12x^2 + 18x + 42]/[ 18 x^2 + 12x +58 ] = [ 2x + 3]/[ 3x + 2] = 42/58`
`therefore [ 2x + 3]/[ 3x + 2] = 42/58`
⇒ `[ 2x + 3]/[ 3x + 2] = 21/29`
⇒ `58x +87 = 63x + 42`
⇒ `63x - 58x = 87 - 42`
⇒ `5x = 45`
⇒ `x = 9`
Thus, the solution of the given equation is x = 9.
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