Question

If \(\alpha\) and \(\beta\) are the roots of the equation \(7\left(\frac{5 x}{3 x-2}\right)-3\left(\frac{3 x-2}{5 x}\right)=4, \frac{1}{\alpha}+\frac{1}{\beta}\) is (a) \(\frac{19}{3}\) (b) \(-\frac{19}{22}\) (c) \(\frac{25}{3}\) (d) \(\frac{19}{22}\)

Short answer

The sum of the reciprocals of the roots \(\frac{1}{\alpha} + \frac{1}{\beta}\) for the given equation is found to be \(\frac{35}{4}\). However, this option is not listed in the choices given.

Step-by-step solution

Simplifying the given equation

To simplify the equation, we have to clear the fractions by eliminating the denominators. For that, we can multiply both sides of the equation by \((3x-2)(5x)\). So we get: \begin{align*} 7\left(\frac{5 x}{3x-2}\right)-3\left(\frac{3 x-2}{5 x}\right)=4 &\implies 7(5x)(3x-2)(5x) - 3(3x-2)(5x)(3x-2) = 4(3x-2)(5x) \\ &\implies 7(15x^2-10x) - 3(9x^2-4) = 20x(3x-2). \end{align*}

Expanding and simplifying the equation

Now, we proceed to expand and simplify the equation to get a quadratic equation: \begin{align*} 7(15x^2-10x)-3(9x^2-4)=20x(3x-2) &\implies 105x^2-70x-27x^2+12 = 60x^2-40x \\ &\implies (105x^2-27x^2) - (70x+40x)+12 = 60x^2-40x \\ &\implies 78x^2-110x+12 = 60x^2-40x \\ &\implies 18x^2-70x+12 = 0. \end{align*}

Applying Vieta's Formulas

From the given quadratic equation \(18x^2-70x+12=0\), we can apply Vieta's Formulas to find \(\frac{1}{\alpha} + \frac{1}{\beta}\). The sum of the roots of a quadratic equation is given by \(\alpha+\beta=-\frac{b}{a}\) and product \(\alpha\beta =\frac{c}{a}\), where \(ax^2+bx+c=0\). So, we get: $$ \alpha+\beta = -\frac{-70}{18} = \frac{35}{9}, $$ and $$ \alpha\beta = \frac{12}{18} = \frac{2}{3}. $$ Now, we can find \(\frac{1}{\alpha} + \frac{1}{\beta}\) using the formula: $$ \frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}. $$ Substituting the values, we get: $$ \frac{1}{\alpha}+\frac{1}{\beta}=\frac{\frac{35}{9}}{\frac{2}{3}}=\frac{35}{6}\times\frac{3}{2}=\frac{35\times3}{6\times2}=\frac{105}{12}. $$

Matching the result to the options given

We have calculated the value of \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{105}{12}\). None of the options match this value exactly, but if we simplify the fraction to lowest terms, then we get: $$ \frac{105}{12} = \frac{35\times3}{4\times3}=\frac{35}{4}. $$ Now, matching our result to the available options, we find that the correct option is not listed.

Key concepts

Vieta's Formulas

Vieta's Formulas provide a very useful relationship between the coefficients of a quadratic equation and the roots of that equation. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the formulas establish that:

• The sum of the roots \(\alpha + \beta = -\frac{b}{a}\).
• The product of the roots \(\alpha \beta = \frac{c}{a}\).

This gives you a simple way to compute the sum and product of the roots without actually solving the quadratic equation by factoring or using the quadratic formula. This is especially useful in cases where knowing the exact roots is not necessary or where finding the roots is time-consuming. By focusing on the sum and product, many problems can be simplified or solved more quickly.

In this case, the quadratic equation from the problem is \(18x^2 - 70x + 12 = 0\). Applying Vieta's formulas, we calculate the sum of the roots as \(\frac{35}{9}\) and the product as \(\frac{2}{3}\). These will come in handy to calculate expressions involving \(\alpha\) and \(\beta\) later, without needing their individual values.

Sum and Product of Roots

Once you have the sum and product of the roots for a quadratic equation, they allow you to easily calculate related expressions like \(\frac{1}{\alpha} + \frac{1}{\beta}\). This particular expression can be determined using the identity \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}\). By substituting in the sum and product from Vieta's formulas, you can quickly find the desired result.

In our exercise, using \(\alpha + \beta = \frac{35}{9}\) and \(\alpha \beta = \frac{2}{3}\), we calculate:

• \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\frac{35}{9}}{\frac{2}{3}} = \frac{35}{6} \times \frac{3}{2} = \frac{105}{12}\).

This nifty algebraic trick saves you from the cumbersome task of finding \(\alpha\) and \(\beta\) directly, and instead works with more straightforward arithmetic to navigate through the options provided in the question.

Fraction Simplification

Fractions are everywhere in math, and simplifying them is a key skill to make expressions easier to handle. Simplification means rewriting the fraction in its simplest form, where the numerator and denominator have no common factors other than 1.

In this problem, after calculating \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{105}{12}\), you simplify the fraction to know if it matches any of the given options. First, check for common factors in the numbers:

• The greatest common divisor (GCD) of 105 and 12 is 3.
• Divide both the numerator and denominator by 3 to get \(\frac{35}{4}\).

However, upon simplification, it's evident that \(\frac{35}{4}\) does not match any of the options originally provided.

Simplifying fractions is not only about reducing numbers but also ensuring they align correctly with given choices or expressions needed in broader equations. Many students find this a crucial step in checking their work or verifying correctness on exams.