Question

plz tell how to find a/b on solving this equation:

(a+b)^2/ab = 4.5

Answer 1

Answer on Question # 42016 – Physics – Other

1. plz tell how to find a/b on solving this equation:

$$(a + b) ^ {2} / a b = 4.5.$$

Solution.

Let open the brackets.

$$\frac {a ^ {2} + 2 a b + b ^ {2}}{a b} = 4.5, \quad \frac {a}{b} + 2 + \frac {b}{a} = 4.5, \quad \frac {a}{b} + \frac {b}{a} = \frac {5}{2}.$$

Let denote $\frac{a}{b} = x$ . The equation takes the form

$$x + \frac {1}{x} = \frac {5}{2}, \quad \frac {x ^ {2} + 1}{x} = \frac {5}{2}, \quad 2 x ^ {2} + 2 = 5 x, \quad 2 x ^ {2} - 5 x + 2 = 0.$$

This is a quadratic equation. The discriminant is $D = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9$ .

So, the roots are $x_{1} = \frac{5 - 3}{2\cdot 2} = \frac{1}{2}, x_{2} = \frac{5 + 3}{2\cdot 2} = 2$ .

Let go back to the initial variables. So, $\frac{a}{b} = \frac{1}{2}$ or $\frac{a}{b} = 2$ .

Answer: $\frac{a}{b} = \frac{1}{2}$ or $\frac{a}{b} = 2$ .

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