Question

solve for x: x square - 14 x + 49 / x square- 49 = 3/ 17​

Answer 1

$\underline{ \underline{ \bold{ \red{ \: \: \: QUESTION\: \: \: }}}}$➡

$\boxed{ \: \bold{ \: \: \: \frac{x {}^{2} - 14x - 49 }{x {}^{2} - 49 \: \: \: } = \frac{3}{17} \: \: \: \: }}$

$\underline{ \underline{ \bold{ \red{ \: \: \:SOLUTION \: \: \: }}}}$➡

$\bold{ \frac{x {}^{2} - 14x + 49 }{x {}^{2} - 49 } = \frac{3}{17} } \\ \\ \\ \bold{ = > \frac{(x) {}^{2} - 2 \times x \times 7 + (7) {}^{2} }{(x) {}^{2} - (7) {}^{2} } = \frac{3}{17} } \\ \\ \\ \bold{ = > \frac{(x - 7) {}^{2} }{(x + 7)(x - 7)} = \frac{3}{17} } \\ \\ \\ \bold{ = > \frac{(x - 7)(x -7 )}{(x + 7)(x - 7)} = \frac{3}{17} } \\ \\ \\ \bold{ = > \frac{x - 7}{x + 7} = \frac{3}{17} } \\ \\ \\ \bold{ = > 17(x - 7) = 3(x + 7)} \\ \\ \\ \bold{ = > 17x - 119 = 3x + 21} \\ \\ \\ \bold{ = > 17x - 3x = 21 + 119} \\ \\ \\ \bold{ = > 14x = 140} \\ \\ \\ \bold{ = > x = \frac{140}{14} } \\ \\ \\ \bold{ = > x = 10}$

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$\underline{ \underline{ \bold{ \red{ \: \: \:VERIFICATION \: \: \: }}}}$➡

$\: \: \bold{ We \: \: have\: ,} \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \bold{ \: \: \: \: x = 10 \: \: \: }} \\ \\ \\ \bold{Putting \: \: the \: \: value \: \: of \: \: \boxed{x}\: \: in \: } \\ \bold{L.H.S. \: \: of \: \: the \: \: equation, \: \: \: we \: \: get,}\\ \\ \\ \bold{L.H.S. = \frac{x {}^{2} - 14x + 49 }{x {}^{2} - 49 } } \\ \\ \\ \bold{ = \frac{(10) {}^{2} - 14 \times (10) + 49}{(10) {}^{2} - 49 } } \\ \\ \\ \bold{ = \frac{100 - 140 + 49}{100 - 49} } \\ \\ \\ \bold{ = \frac{9}{51} } \\ \\ \\ \bold{ = \frac{3 \times 3}{17 \times 3} } \\ \\ \\ \bold{ = \frac{3}{17} } \\ \\ \\ \bold{ = R.H.S.} \\ \\ \\ \\ \bold{So, \: \: \: \: \: \: \: \: \: \: \: \underline{ \: \: \: L.H.S.= R.H.S. \: \: \: }}$



$\bold{\: Hence \:,}$


$\boxed{\boxed{\pink{\bold{\: \: The \: \: value \: \: of \: \: x \: = \: 10 \: \:}}}}$

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✝$\mathfrak{\purple{THANKS..}}$✝

Answer 2

$\underline{\mathfrak{\huge{Question:}}}$

Solve for the value of x, in the equation :-

$\frac{x^{2} - 14x + 49}{x^{2} - 49}$

$\underline{\mathfrak{\huge{Answer:}}}$

●First, we will start with factorising and simplifying the L.H.S.

●We can do so by applying the splitting middle term method in the two equations given to us in the L.H.S. :-

=》 $\frac{x^{2} - 14x + 49}{x^{2} - 7^{2}}$

=》 $\frac{x^{2} - 7x - 7x + 49}{x^{2} - 7^{2}}$

=》 $\frac{x ( x - 7 ) - 7 ( x - 7 )}{( x - 7 ) ( x + 7 )}$

=》 $\frac{( x - 7 ) ( x - 7 )}{( x - 7 ) ( x + 7 )}$

=》 $\frac{( x - 7 )}{( x + 7 )}$

When we keep R.H.S. = L.H.S., we get the result as :-

=》 $\frac{( x - 7 )}{( x + 7 )} = \frac{3}{17}$

By doing the necessary cross multiplying, we get the result to be :-

=》 3 ( x + 7 ) = 17 ( x - 7 )

=》 3x + 21 = 17x - 119

=》 21 + 119 = 17x - 3x

=》 140 = 14x

=》 x = $\frac{140}{14}$

=》 $\boxed{\tt{x = 10}}$

The Verification of the equation :-

$\frac{x^{2} - 14x + 49}{x^{2} - 49}$

=》 $\frac{10^{2} - 14(10) + 49}{10^{2} - 49}$

=》 $\frac{100 - 140 + 49}{100 - 49}$

=》 $\frac{9}{51}$

=》 $\tt{\frac{3}{17}}$

Now that we've got :-

R.H.S. = L.H.S.

$\underline{\tt{Hence\:Proved!!}}$