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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

All the values of \(\mathrm{m}\) for which both roots of the equation \(\mathrm{x}^{2}-2 \mathrm{mx}+\mathrm{m}^{2}-1=0\) are greater then \(-2\) but less than 4 lie in interval (a) \(\mathrm{m}<3\) (b) \(-1>\mathrm{m}<3\) (c) \(1<\mathrm{m}<4\) (d) \(-2<\mathrm{m}<\mathrm{o}\)

If the roots of the equation \(b x^{2}+c x+a=0\) be imaginary then for all real values of \(x\) the expression \(3 b^{2} x^{2}+6 b c x+2 c^{2}\) is (a) \(<4 \mathrm{ab}\) (b) \(>-4 \mathrm{ab}\) (c) \(-4 \mathrm{ab}\) (d) \(>4 \mathrm{ab}\)

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) \(3 x^{2}-25 x+3=0\)

If one root of the equation \(4 \mathrm{x}^{2}-6 \mathrm{x}+\mathrm{p}=0\) is \(\mathrm{q}+2 \mathrm{i}\), where \(\mathrm{p}, \mathrm{q} \in \mathrm{R}\) then \(\mathrm{p}+\mathrm{q}=\) (a) 10 (b) 19 (c) \(-24\) (d) \(-32\)

For which value of \(b\), equation \(x^{2}+b x-1=0\) and \(\mathrm{x}^{2}+\mathrm{x}+\mathrm{b}=0\) have a common root. (a) \(-\sqrt{2}\) (b) \(-\mathrm{i} \sqrt{3}\) (c) \(\mathrm{i} \sqrt{5}\) (d) \(\sqrt{2}\)
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