If the roots of the equation 5x² - 7x + k = 0 are reciprocal to each other, then what is the value of k?
Understand the Problem
The question is asking us to determine the value of k in the quadratic equation 5x² - 7x + k = 0, given that the roots of the equation are reciprocals of each other. This involves using the properties of quadratic equations and relationships between the coefficients and roots.
Answer
The value of \(k\) is \(5\).
Answer for screen readers
The value of (k) is (5).
Steps to Solve
- Understanding Reciprocal Roots
If the roots of the quadratic equation (5x^2 - 7x + k = 0) are reciprocal to each other, then if one root is (r), the other root must be (\frac{1}{r}).
- Using Vieta's Formulas
From Vieta's formulas, the sum of the roots is given by: $$ r + \frac{1}{r} = \frac{-b}{a} = \frac{7}{5} $$ This is because (a = 5) and (b = -7).
- Setting Up the Equation
The product of the roots can be expressed as: $$ r \cdot \frac{1}{r} = 1 = \frac{c}{a} = \frac{k}{5} $$ This implies: $$ k = 5 $$
- Final Value of k
Thus, the value of (k) is directly computed from the product.
The value of (k) is (5).
More Information
The relationship between roots and coefficients in a quadratic equation is fundamental in algebra. When the roots are reciprocals, it leads directly to specific values for the coefficients. This method can be applied to similar problems with different quadratic equations.
Tips
- Incorrectly applying Vieta’s formulas; ensure to use the correct relationships for sum and product of roots.
- Misunderstanding reciprocal roots; remember that if one root is (r), the other must be (\frac{1}{r}).
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