Question

Math 9 Learning Task 2; A.translate the following verbal sentences to mathematical sentence.then express into quadratic equations in terms of "x" B.Solve for x: $\frac{1}{x} + \frac{1}{x + 10} = \frac{1}{10}$ ​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{\dfrac{1}{x}+\dfrac{1}{x+10}=\dfrac{1}{10}}$

$\underline{\textsf{To find:}}$

$\textsf{Roots of the equation}$

$\underline{\textsf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{1}{x}+\dfrac{1}{x+10}=\dfrac{1}{10}}$

$\mathsf{\dfrac{x+10+x}{x(x+10)}=\dfrac{1}{10}}$

$\mathsf{\dfrac{2x+10}{x^2+10x}=\dfrac{1}{10}}$

$\mathsf{10(2x+10)=x^2+10x}$

$\mathsf{20x+100=x^2+10x}$

$\implies\boxed{\mathsf{x^2-10x-100=0}}$

$\textsf{We apply quadratic formula to find  the roots of the equation}$

$\boxed{\mathsf{x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}}}$

$\implies\mathsf{x=\dfrac{10\pm\sqrt{10^2-4{\times}1{\times}(-100)}}{2{\times}1}}$

$\implies\mathsf{x=\dfrac{10\pm\sqrt{100+400}}{2}}$

$\implies\mathsf{x=\dfrac{10\pm\sqrt{500}}{2}}$

$\implies\mathsf{x=\dfrac{10\pm\sqrt{100{\times}5}}{2}}$

$\implies\mathsf{x=\dfrac{10\pm10\sqrt{5}}{2}}$

$\implies\mathsf{x=5\pm5\sqrt{5}}$

$\therefore\mathsf{\bf\,Roots\;are\;5+5\sqrt{5}\;\;\&\;\;5-5\sqrt{5}}$

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