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3 years ago

How to solve the rational equation? 1/3 + 1/2 = 1/x

Mathematics

1 answer:

tatyana61 [14]3 years ago

8 0

Answer:

x = ⁶/₅

Step-by-step explanation:

$\begin{array}{rcll}\dfrac{1}{3} + \dfrac{1}{2} & = &\dfrac{1}{x} & \\\\2x + 3x & = & 6 & \text{Multiplied all terms by LCM of denominators (6x)}\\5x & = & 6 & \text{Simplified left-hand side}\\x & = & \mathbf{\dfrac{6}{5}} & \text{Divided each side by 5}\\\end{array}$

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$use\ Vieta's\ formula:\\\\\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}=\frac{\alpha^2+2\alpha\beta+\beta^2-2\alpha\beta}{\alpha\beta}=\frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}=\frac{(\alpha+\beta)^2}{\alpha\beta}-2\\\\\alpha+\beta=\frac{-b}{a};\ \alpha\beta=\frac{c}{a}\\\\\frac{(\alpha+\beta)^2}{\alpha\beta}-2=\frac{\left(\frac{-b}{a}\right)^2}{\frac{c}{a}}-2=\frac{b^2}{a^2}\cdot\frac{a}{c}-2=\frac{b^2}{ac}-2$

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