Question

If \(\alpha \& \beta\) are the roots of the equation \(x^{2}-x+1=0\) then \(\alpha^{2009}+\beta^{2009}=\) (a) \(-1\) (b) 1 (c) \(-2\) (d) 2

Short answer

The value of \(\alpha^{2009}+\beta^{2009}\) is 1. Therefore, the correct answer is (b) 1.

Step-by-step solution

Write down Vieta's Formulas for quadratic equations

Vieta's Formulas for a quadratic equation \(ax^2+bx+c=0\) with roots \(\alpha\) and \(\beta\) are: 1. \(\alpha+\beta=-\frac{b}{a}\) 2. \(\alpha\cdot\beta=\frac{c}{a}\)

Apply Vieta's Formulas to the given equation

For the given equation \(x^2-x+1=0\) (where \(a=1\), \(b=-1\), \(c=1\)), apply Vieta's Formulas: 1. \(\alpha+\beta=-\frac{(-1)}{1}=1\) 2. \(\alpha\cdot\beta=\frac{1}{1}=1\)

Find a relationship between \(\alpha^2, \beta^2\) and \(\alpha, \beta\)

Notice that: \[(\alpha+\beta)^2 = 1^2 = 1\] Expanding this, we get: \[\alpha^2 + 2\alpha\beta + \beta^2 = 1\] Since \(\alpha\beta = 1\), we have: \[\alpha^2 + \beta^2 = 1 - 2\] Therefore, \(\alpha^2+\beta^2 = -1\)

Raise the relationship found in step 3 to the power 1004 and multiply by the sum of roots

To find \(\alpha^{2009}+\beta^{2009}\), we can manipulate the relationship found in step 3 and use exponent properties: \[(\alpha^2+\beta^2)^{1004} = (-1)^{1004}= 1\] Now, multiply by \(\alpha+\beta=1\) to get: \[\alpha^{2009}+\beta^{2009} = (\alpha^2+\beta^2)^{1004}(\alpha+\beta) = 1 \cdot 1 = 1\]

Write the final answer

The value of \(\alpha^{2009}+\beta^{2009}\) is 1. Therefore, the correct answer is (b) 1.

Key concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra that students often encounter. These equations are characterized by a standard form, given by the equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The solutions to these equations are often referred to as the 'roots' and they can be real or complex numbers.

The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of the coefficient \(a\). To find the roots of a quadratic equation, one may use various methods such as factoring, completing the square, or applying the quadratic formula. Understanding the nature of these solutions is crucial because it can provide insights into the properties and behavior of the function represented by the equation.

Vieta's Formulas in Quadratic Equations

Vieta's Formulas offer a direct relationship between the roots of a quadratic equation and its coefficients. The beauty of these formulas lies in their ability to relate the sum and product of the roots directly to the coefficients of the equation, providing an elegant link between the algebraic expressions and the roots without the need to solve the equation explicitly.

Relationship Between Roots

The roots of a quadratic equation, typically denoted as \(\alpha\) and \(\beta\), have a fascinating relationship with each other when examined through the lens of Vieta's Formulas. According to Vieta, the sum of the roots \(\alpha + \beta\) equals \(-\frac{b}{a}\) and their product \(\alpha\beta\) equals \(\frac{c}{a}\), where \(a\), \(b\), and \(c\) are the coefficients from the standard form of the quadratic equation.

These relationships can be particularly powerful when working with higher powers of the roots, as in the given exercise. By understanding the sum and product of the roots, one can manipulate these quantities when the roots themselves are raised to high powers, often simplifying what might seem at first glance to be a daunting arithmetic task.

Combining Sum and Product of Roots

When roots are involved in arithmetic operations, we can apply the relationship between roots to simplify the expressions. For instance, expressions like \(\alpha^2 + \beta^2\) and \(\alpha^3 + \beta^3\) can be derived from the sum and product of \(\alpha\) and \(\beta\). These relationships are invaluable when solving complex problems involving roots of quadratic equations.

Exponent Properties

Exponents represent repeated multiplication and follow specific laws that make calculations with powers more manageable. Some of the most essential exponent properties include the product of powers, power of a power, and power of a product.

The product of powers property states that when multiplying two exponents with the same base, you add the exponents. In contrast, the power of a power property indicates that when raising an exponent to another power, you multiply the exponents. These properties are critical in simplifying expressions where variables are raised to significant powers, as seen in our exercise.

Applying Exponent Properties to Roots

When using exponent properties in the context of the roots of a quadratic equation, one can evaluate expressions involving high powers of these roots without actual multiplication. This technique was demonstrated in the original solution by raising \(\alpha^2 + \beta^2\) to the 1004th power, effectively using the power of a power property to simplify the complex expression \(\alpha^{2009} + \beta^{2009}\), showcasing the profound impact of these properties on solving equations.