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Chapter 3: Problem 173

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

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

Step by step solution

01

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.
02

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 \(\alpha^{2} + \beta^{2}\).
03

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.

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Most popular questions from this chapter

For the equation \(\mathrm{x}^{2}+\mathrm{k}^{2}=(2 \mathrm{k}+2) \mathrm{x}, \mathrm{k} \in \mathrm{R}\) roots are complex then (a) \(\overline{k=\\{(-1) / 2\\}}\) (b) \(k>\\{(-1) / 2\\}\) (c) \(k>\\{(-1) / 2\\}\) (d) \(\\{(-1) / 2\\}<\mathrm{k}<0\)

If the sum of the two roots of the equation \([1 /(x+a)]+[1 /(x+b)]=(1 / k)\) is zero then their Product is (a) \(\overline{(1 / 2)\left(a^{2}+b^{2}\right)}\) (b) \(\\{(-1) / 2\\}(a+b)^{2}\) (c) \([\\{a+b\\} / 2]^{2}\) (d) None

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

If \(\mathrm{b}^{2}=\) ac, equation \(\mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) and \(\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0\) have common roots then \((\mathrm{d} / \mathrm{a}),(\mathrm{e} / \mathrm{b}),(\mathrm{f} / \mathrm{c})\) are in (a) A.P. (b) G.P. (c) H.P. (d) None of these

The sum of the roots of the equation \(\mathrm{x}^{2}-3|\mathrm{x}|-10=0\) is (a) 3 (b) \(-3\) (c) \(-10\) (d) 0
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