求根公式(Latex版)

文章目录

  • 一次方程的求根公式
  • 二次方程的求根公式
  • 三次方程的求根公式
  • 四次方程的求根公式

The Quartic Formula

一次方程的求根公式

x = − b a x = {-b \over a} x=ab

The linear formula gives the solution of a x + b = 0 ax+b=0 ax+b=0 for real numbers a a a, b b b with a ≠ 0 a\neq0 a=0.

二次方程的求根公式

x = − b ± b 2 − 4 a c 2 a x = {-b\pm\sqrt{b^2-4ac} \over 2a} x=2ab±b24ac

The quadratic formula gives the solutions of a x 2 + b x + c = 0 ax^2+bx+c=0 ax2+bx+c=0 for real numbers a a a, b b b, c c c with a ≠ 0 a\neq0 a=0.

三次方程的求根公式

x = − 2 b + ( − 1 + − 3 2 ) n 4 ( − 2 b 3 + 9 a b c − 27 a 2 d + ( − 2 b 3 + 9 a b c − 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ) 3 + ( − 1 − − 3 2 ) n 4 ( − 2 b 3 + 9 a b c − 27 a 2 d − ( − 2 b 3 + 9 a b c − 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ) 3 6 a x = {-2b + \bigl({-1+\sqrt{-3}\over2}\bigr)^n\sqrt[3]{4\bigl(-2b^3+9abc-27a^2d+\sqrt{(-2b^3+9abc-27a^2d)^2-4(b^2-3ac)^3}\bigr)}+\bigl({-1-\sqrt{-3}\over2}\bigr)^n\sqrt[3]{4\bigl(-2b^3+9abc-27a^2d-\sqrt{(-2b^3+9abc-27a^2d)^2-4(b^2-3ac)^3}\bigr)} \over 6a} x=6a2b+(21+3 )n34(2b3+9abc27a2d+(2b3+9abc27a2d)24(b23ac)3 ) +(213 )n34(2b3+9abc27a2d(2b3+9abc27a2d)24(b23ac)3 )

The cubic formula gives the solutions of a x 3 + b x 2 + c x + d = 0 ax^3+bx^2+cx+d=0 ax3+bx2+cx+d=0 for real numbers a a a, b b b, c c c, d d d with a ≠ 0 a\neq0 a=0.

Directions: Take n = 0 n=0 n=0, 1 1 1, 2 2 2. Use real cube roots if possible, and principal roots otherwise.

四次方程的求根公式

x = − 3 b ± ( 3 ( 3 b 2 − 8 a c + 2 a 4 ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 ) 3 + 2 a 4 ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e − ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 ) 3 ) ± 3 ( 3 b 2 − 8 a c + 2 a ( − 1 + − 3 2 ) 4 ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 ) 3 + 2 a ( − 1 − − 3 2 ) 4 ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e − ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 ) 3 ) ) ± s g n ( ( s g n ( − b 3 + 4 a b c − 8 a 2 d ) − 1 2 ) ( s g n ( max ⁡ ( ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 , min ⁡ ( 3 b 2 − 8 a c , 3 b 4 + 16 a 2 c 2 + 16 a 2 b d − 16 a b 2 c − 64 a 3 e ) ) ) − 1 2 ) ) 3 ( 3 b 2 − 8 a c + 2 a ( − 1 − − 3 2 ) 4 ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 ) 3 + 2 a ( − 1 + − 3 2 ) 4 ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e − ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 ) 3 ) 12 a \def\sgn{\mathop{\rm sgn}} x = {-3b\pm\biggl(\sqrt{3\Bigl(3b^2-8ac+2a\sqrt[3]{4\bigl(2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}\bigr)}+2a\sqrt[3]{4\bigl(2c^3-9bcd+27ad^2+27b^2e-72ace-\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}\bigr)}\Bigr)}\pm\sqrt{3\Bigl(3b^2-8ac+2a\bigl({-1+\sqrt{-3}\over2}\bigr)\sqrt[3]{4\bigl(2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}\bigr)}+2a\bigl({-1-\sqrt{-3}\over2}\bigr)\sqrt[3]{4\bigl(2c^3-9bcd+27ad^2+27b^2e-72ace-\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}\bigr)}\Bigr)}\biggr)\pm\sgn\biggl(\Bigl(\sgn\bigl(-b^3+4abc-8a^2d\bigr)-{1\over2}\Bigr)\Bigl(\sgn\bigl(\max((2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3,\min(3b^2-8ac,3b^4+16a^2c^2+16a^2bd-16ab^2c-64a^3e))\bigr)-{1\over2}\Bigr)\biggr)\sqrt{3\Bigl(3b^2-8ac+2a\bigl({-1-\sqrt{-3}\over2}\bigr)\sqrt[3]{4\bigl(2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}\bigr)}+2a\bigl({-1+\sqrt{-3}\over2}\bigr)\sqrt[3]{4\bigl(2c^3-9bcd+27ad^2+27b^2e-72ace-\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}\bigr)}\Bigr)}\over12a} x=12a3b±(3(3b28ac+2a34(2c39bcd+27ad2+27b2e72ace+(2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3 ) +2a34(2c39bcd+27ad2+27b2e72ace(2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3 ) ) ±3(3b28ac+2a(21+3 )34(2c39bcd+27ad2+27b2e72ace+(2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3 ) +2a(213 )34(2c39bcd+27ad2+27b2e72ace(2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3 ) ) )±sgn((sgn(b3+4abc8a2d)21)(sgn(max((2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3,min(3b28ac,3b4+16a2c2+16a2bd16ab2c64a3e)))21))3(3b28ac+2a(213 )34(2c39bcd+27ad2+27b2e72ace+(2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3 ) +2a(21+3 )34(2c39bcd+27ad2+27b2e72ace(2c39bcd+27ad2+27b2e72ace)24(c23bd+12ae)3 ) )

The quartic formula gives the solutions of a x 4 + b x 3 + c x 2 + d x + e = 0 ax^4+bx^3+cx^2+dx+e=0 ax4+bx3+cx2+dx+e=0 for real numbers a a a, b b b, c c c, d d d, e e e with a ≠ 0 a\neq0 a=0.

Directions: Choose all possibilities for the three ± \pm ± signs with the last two equivalent. Use real cube roots if possible, and principal roots otherwise.

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