messages listlengths 1 1 | ground_truth stringlengths 33 79 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "x^5 + 10x^3 + 20x - 4 = 0. Find a closed form expression for the real roots of x in the simplest form.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2^{\frac{3}{5}} - 2^{\frac{2}{5}} | MATH |
[
{
"content": "x^5 + 5x^3 + 5x - 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed... | (1 + \sqrt{2})^{\frac{1}{5}} + -1(1 + \sqrt{2})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 10x^3 + 20x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (-14 + 2\sqrt{57})^{\frac{1}{5}} + -2(-14 + 2\sqrt{57})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 15x^3 + 45x - 26 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (13 + 2\sqrt{103})^{\frac{1}{5}} + -3(13 + 2\sqrt{103})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 20x^3 + 80x - 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (33 + \sqrt{2113})^{\frac{1}{5}} + -4(33 + \sqrt{2113})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 30x^3 + 180x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\... | (61 + \sqrt{11497})^{\frac{1}{5}} + -6(61 + \sqrt{11497})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 - 10x^3 + 20x + 36 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (-18 + 2\sqrt{73})^{\frac{1}{5}} + 2(-18 + 2\sqrt{73})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 5x^3 + 5x - 30 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxe... | (15 + \sqrt{226})^{\frac{1}{5}} + -1(15 + \sqrt{226})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 40x^3 + 320x - 258 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\... | (129 + \sqrt{49409})^{\frac{1}{5}} + -8(129 + \sqrt{49409})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 15x^3 + 45x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (-14 + \sqrt{439})^{\frac{1}{5}} + -3(-14 + \sqrt{439})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 - 5x^3 + 5x - 6 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed... | (3 + 2\sqrt{2})^{\frac{1}{5}} + -1(3 + 2\sqrt{2})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 25x^3 + 125x - 126 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\... | (63 + \sqrt{7094})^{\frac{1}{5}} + -5(63 + \sqrt{7094})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 10x^3 + 20x + 56 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (-28 + 4\sqrt{51})^{\frac{1}{5}} + -2(-28 + 4\sqrt{51})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 5x^3 + 5x + 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed... | (-1 + \sqrt{2})^{\frac{1}{5}} + -1(-1 + \sqrt{2})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 30x^3 + 180x - 344 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\... | (172 + 4\sqrt{2335})^{\frac{1}{5}} + -6(172 + 4\sqrt{2335})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 - 20x^3 + 80x + 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\bo... | (-33 + \sqrt{65})^{\frac{1}{5}} + -4(-33 + \sqrt{65})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 5x^3 + 5x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\box... | (61 + \sqrt{3722})^{\frac{1}{5}} + -1(61 + \sqrt{3722})^{-\frac{1}{5}} | MATH |
[
{
"content": "x^5 + 50x^3 + 500x - 1252 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \... | (626 + 2\sqrt{122969})^{\frac{1}{5}} + -10(626 + 2\sqrt{122969})^{-\frac{1}{5}} | MATH |
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