Question

If \(\alpha+\beta=5 \alpha^{2}=5 \alpha-3\) and \(\beta^{2}=5 \beta-3\) then the equation whose roots are \((\alpha / \beta)\) and \((\beta / \alpha)\) is (a) \(3 x^{2}-19 x+3=0\) (b) \(x^{2}+5 x-3=0\) (c) \(x^{2}-5 x+3=0\) (d) \(3 x^{2}-25 x+3=0\)

Short answer

The equation whose roots are \((\alpha / \beta)\) and \((\beta / \alpha)\) is \((a) 3x^2 - 19x + 3 = 0\).

Step-by-step solution

Find the sum and product of the reciprocal roots

Let \(p = \frac{\alpha}{\beta}\) and \(q = \frac{\beta}{\alpha}\). We need to find \(p + q\) and \(pq\). Adding both equations, we get \(p + q = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}\). Next, we find the product of \(p\) and \(q\): \( pq = \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1\).

Substitute known expressions for \(\alpha^2\) and \(\beta^2\)

From the givens, we have: \(\alpha^2 = 5\alpha - 3\) \(\beta^2 = 5\beta - 3\) Now, we substitute \(\alpha^2\) and \(\beta^2\) in our expression of \(p + q\) which is \(\frac{\alpha^2 + \beta^2}{\alpha\beta}\): \(p + q = \frac{(5\alpha - 3) + (5\beta - 3)}{\alpha\beta} = \frac{5(\alpha + \beta) - 6}{\alpha\beta}\)

Substitute value of \(\alpha + \beta\)

According to the exercise, \(\alpha+\beta=5\). So we can substitute this value into the equation we found in step 2: \(p + q = \frac{5(5) - 6}{\alpha\beta} = \frac{19}{\alpha\beta}\)

Vieta's formulas

Now that, we've found \(p + q\) and \(pq\), we can use Vieta's formulas to derive the corresponding quadratic equation. If \(p\) and \(q\) are the roots of the quadratic equation, then the equation can be expressed as \(x^2 - (p + q)x + pq\). Let's plug in the values of \(p + q\) and \(pq\): \(x^2 - (\frac{19}{\alpha\beta})x + 1 = 0\) To change the denominator from \(\alpha\beta\) to an integer, we multiply the whole equation by \(\alpha\beta\): \(\alpha\beta x^2 - 19x + \alpha\beta = 0\)

Conclusion

Therefore, the equation whose roots are \((\alpha / \beta)\) and \((\beta / \alpha)\) is \(\alpha\beta x^2 - 19x + \alpha\beta = 0\). Looking at the choices, we can see that the correct answer is \((a) 3x^2 - 19x + 3 = 0\) (since it resembles the equation we derived, but both the leading and constant coefficients have been divided by the actual value of \(\alpha\beta\)).

Key concepts

Vieta's Formulas

Vieta's formulas provide a straightforward way to relate the coefficients of a polynomial to sums and products of its roots. They are especially useful for quadratic equations. For a standard quadratic equation, \[ax^2 + bx + c = 0\]Vieta's formulas tell us:

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

These relationships allow quick insight into the nature of the roots without finding them explicitly. In this exercise, Vieta's formulas are used to frame the quadratic equation whose roots are expressed as ratios (\(\alpha/\beta\) and \(\beta/\alpha\)).
By substituting the derived value of these expressions into the standard form, we achieve a modified equation. Multiplying through by \(\alpha\beta\) simplifies it, and matches one of the given multiple-choice answers.

Roots of Equations

In algebra, roots of an equation refer to the values of the variable that satisfy the equation. For quadratic equations, the goal is to find two numbers (roots) that after substitution into the equation make it equal zero.
Roots can be found using various methods such as factoring, completing the square, or using the quadratic formula.
For this specific exercise, the roots \((p, q)\) are given as ratios of other unknown variables, \(\alpha/\beta\) and \(\beta/\alpha\). These reciprocal roots must adhere to the original equation derived through substitution and simplification.

Sum and Product of Roots

Understanding the sum and product of the roots provides a fundamental grasp of polynomial structure. They directly reflect the relationships derived from Vieta's formulas. For the quadratic equation given as \[ax^2 + bx + c = 0,\]the expressions for the sum and product of the roots \((\alpha, \beta)\) are:

• Sum: \(\alpha + \beta = -\frac{b}{a}\)
• Product: \(\alpha\beta = \frac{c}{a}\)

When solving the exercise, these relationships help in determining unknown variables. In this problem, knowing \(\alpha + \beta = 5\) simplifies the steps toward finding \(p + q\).
By aligning these with the derived expressions and transforming them into useful forms (like multipliers of \(\alpha\beta\)), we can solve for other parts of the equation effectively.