Question

If \(\alpha \& \beta\) are roots of quadratic equation \(x^{2}+13 x+8=0\) then the value of \(\alpha^{4}+\beta^{4}=\) (a) 23281 (b) 23218 (c) 23128 (d) 23182

Short answer

The value of \(\alpha^4 + \beta^4\) is 23281.

Step-by-step solution

Identifying the coefficients of the quadratic equation and using them to compute the value of roots' sum and product

From the given quadratic equation \(x^{2}+13x+8 = 0\), it can be seen that \(a = 1\), \(b = 13\), and \(c = 8\). Let's plug these values into \(\alpha + \beta = -\frac{b}{a}\) and \(\alpha\beta = \frac{c}{a}\) to find \(\alpha + \beta\) and \(\alpha\beta\) respectively.

Computing the value of \(\alpha^2 + \beta^2\)

The value of \(\alpha^{2} + \beta^{2}\) can be found by squaring \(\alpha + \beta\) and subtracting 2 times \(\alpha\beta\) from it. Recall from the previous step that we've already found the values of \(\alpha + \beta\) and \(\alpha\beta\), so let's substitute these into the equation to obtain \(\alpha^{2} + \beta^{2}\).

Computing the value of \(\alpha^4 + \beta^4\)

The value of \(\alpha^4 + \beta^4\) can be found using the equation \((\alpha^{2} + \beta^{2})^{2} - 2 \cdot (\alpha\beta)^{2}\). Again, substituting the values of \(\alpha^{2} + \beta^{2}\) and \(\alpha\beta\) found in the previous steps into this equation will yield the desired \(\alpha^4 + \beta^4\). After carrying out the calculations in the aforementioned steps, compare the calculated value of \(\alpha^{4} + \beta^{4}\) with the given choices to find the correct answer.

Key concepts

Roots of Equation

In the context of quadratic equations, the "roots" are the solutions to the equation. For a quadratic equation in the form of \( ax^2 + bx + c = 0 \), the roots can be denoted as \( \alpha \) and \( \beta \). These roots can be found using the quadratic formula:

• \( \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

However, in some cases, you don't need to calculate the roots explicitly to solve a problem. Various properties and relationships of the roots can often be used instead. This is particularly useful when asked to compute expressions involving higher powers of the roots, like \( \alpha^4 + \beta^4 \). Understanding how to maneuver these relationships and properties without directly solving for \( \alpha \) and \( \beta \) is a key skill when working with quadratic equations.

Sum and Product of Roots

The sum and product of the roots of a quadratic equation is a well-known property derived directly from the coefficients of the equation. According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \), the following relationships hold:

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

For our specific equation \( x^2 + 13x + 8 = 0 \), substituting the values of \( a = 1 \), \( b = 13 \), and \( c = 8 \) gives:

• \( \alpha + \beta = -13 \)
• \( \alpha \beta = 8 \)

These relationships are essential as they allow us to express other polynomial functions of the roots without actually knowing their specific numerical values.

Powers of Roots

Finding higher powers of the roots, such as \( \alpha^4 + \beta^4 \), can seem challenging without knowing the roots themselves. However, with the sums and products of the roots, we can use elegant algebraic identities to achieve this. One such identity involves expressing \( \alpha^2 + \beta^2 \) as:

• \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \)

For our example, this results in:

• \( \alpha^2 + \beta^2 = (-13)^2 - 2 \times 8 = 169 - 16 = 153 \)

Once we have \( \alpha^2 + \beta^2 \), we can find \( \alpha^4 + \beta^4 \) using the identity:

• \( \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2 \)
• Substituting the known values, we get:\
• \( 153^2 - 2 \times 64 = 23409 - 128 = 23281 \)

Therefore, the solution to our initial problem is 23281.

Vieta's Formulas

Vieta's formulas are a powerful set of equations that relate the coefficients of a polynomial to sums and products of its roots. In particular, for quadratic equations, they provide a straightforward way to calculate the sum and product of the roots based on the polynomial's coefficients.These formulas are given as:

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

Using these formulas, we can derive complex expressions involving roots without the need to solve the equation directly. For example, in the exercise given, we used Vieta's formulas to derive \( \alpha^4 + \beta^4 \) by finding the intermediate terms like \( \alpha^2 + \beta^2 \), and using identities for powers of roots.Mastering Vieta’s formulas can greatly simplify your work with polynomials, especially when direct root calculations are unwieldy or unnecessary.