To solve for the equation whose roots are the reciprocals of the roots of the quadratic equation [tex]\( 5x^2 - x + 2 = 0 \)[/tex], let us perform the following steps:
1. Start with the given quadratic equation:
[tex]\[
5x^2 - x + 2 = 0
\][/tex]
2. Let the roots of this equation be [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]. We need to find the equation whose roots are [tex]\( \frac{1}{\alpha} \)[/tex] and [tex]\( \frac{1}{\beta} \)[/tex].
3. For a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], if the roots are [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex], then the sum of the roots [tex]\(\alpha + \beta\)[/tex] is given by [tex]\(-\frac{b}{a}\)[/tex] and the product of the roots [tex]\(\alpha \beta\)[/tex] is given by [tex]\(\frac{c}{a}\)[/tex].
4. By reciprocal property, if [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are the roots of equation [tex]\( 5x^2 - x + 2 = 0 \)[/tex], then [tex]\( \frac{1}{\alpha} \)[/tex] and [tex]\( \frac{1}{\beta} \)[/tex] will be the roots of the new equation. The new coefficients will be formed by swapping [tex]\(a\)[/tex] and [tex]\(c\)[/tex] and keeping [tex]\(b\)[/tex] as it is:
[tex]\[
cx^2 + bx + a = 0
\][/tex]
5. Let's rewrite the original equation, [tex]\(5x^2 - x + 2\)[/tex], in terms of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
ax^2 + bx + c = 0
\][/tex]
where [tex]\(a = 5\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = 2\)[/tex].
6. The new quadratic equation, where the roots will be reciprocals [tex]\( \frac{1}{\alpha} \)[/tex] and [tex]\( \frac{1}{\beta} \)[/tex], will then be:
[tex]\[
2x^2 - x + 5 = 0
\][/tex]
Thus, the equation whose roots are the reciprocals of the roots of [tex]\( 5x^2 - x + 2 = 0 \)[/tex] is:
[tex]\[
2x^2 - x + 5 = 0
\][/tex]
This corresponds to the coefficients [tex]\((2, -1, 5)\)[/tex]. Therefore, the correct answer is:
None of the given options match the correct equation, so the answer is (E) None.
