Lahenda (x+t)^3-x^3=3t(x+1)^2

$$( x + t ) ^ { 3 } - x ^ { 3 } = 3 t ( x + 1 ) ^ { 2 }$$

Lahendage ja leidke x (complex solution)

$$\left\{\begin{matrix}x=-\frac{t^{2}-3}{3\left(t-2\right)}\text{, }&t\neq 2\\x\in \mathrm{C}\text{, }&t=0\end{matrix}\right,$$

Juhised lineaarvõrrandi lahendamiseks

Kasutage kaksliikme \left(x+t\right)^{3} arendamiseks binoomvalemit \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}.

$$x^{3}+3x^{2}t+3xt^{2}+t^{3}-x^{3}=3t\left(x+1\right)^{2}$$

Kombineerige x^{3} ja -x^{3}, et leida 0.

$$3x^{2}t+3xt^{2}+t^{3}=3t\left(x+1\right)^{2}$$

Kasutage kaksliikme \left(x+1\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$3x^{2}t+3xt^{2}+t^{3}=3t\left(x^{2}+2x+1\right)$$

Kasutage distributiivsusomadust, et korrutada 3t ja x^{2}+2x+1.

$$3x^{2}t+3xt^{2}+t^{3}=3tx^{2}+6tx+3t$$

Lahutage mõlemast poolest 3tx^{2}.

$$3x^{2}t+3xt^{2}+t^{3}-3tx^{2}=6tx+3t$$

Kombineerige 3x^{2}t ja -3tx^{2}, et leida 0.

$$3xt^{2}+t^{3}=6tx+3t$$

Lahutage mõlemast poolest 6tx.

$$3xt^{2}+t^{3}-6tx=3t$$

Lahutage mõlemast poolest t^{3}.

$$3xt^{2}-6tx=3t-t^{3}$$

Kombineerige kõik liikmed, mis sisaldavad x.

$$\left(3t^{2}-6t\right)x=3t-t^{3}$$

Jagage mõlemad pooled 3t^{2}-6t-ga.

$$\frac{\left(3t^{2}-6t\right)x}{3t^{2}-6t}=\frac{t\left(3-t^{2}\right)}{3t^{2}-6t}$$

3t^{2}-6t-ga jagamine võtab 3t^{2}-6t-ga korrutamise tagasi.

$$x=\frac{t\left(3-t^{2}\right)}{3t^{2}-6t}$$

Jagage t\left(3-t^{2}\right) väärtusega 3t^{2}-6t.

$$x=\frac{3-t^{2}}{3\left(t-2\right)}$$

Lahendage ja leidke x

$$\left\{\begin{matrix}x=-\frac{t^{2}-3}{3\left(t-2\right)}\text{, }&t\neq 2\\x\in \mathrm{R}\text{, }&t=0\end{matrix}\right,$$

Juhised lineaarvõrrandi lahendamiseks

Kasutage kaksliikme \left(x+t\right)^{3} arendamiseks binoomvalemit \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}.

$$x^{3}+3x^{2}t+3xt^{2}+t^{3}-x^{3}=3t\left(x+1\right)^{2}$$

Kombineerige x^{3} ja -x^{3}, et leida 0.

$$3x^{2}t+3xt^{2}+t^{3}=3t\left(x+1\right)^{2}$$

Kasutage kaksliikme \left(x+1\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$3x^{2}t+3xt^{2}+t^{3}=3t\left(x^{2}+2x+1\right)$$

Kasutage distributiivsusomadust, et korrutada 3t ja x^{2}+2x+1.

$$3x^{2}t+3xt^{2}+t^{3}=3tx^{2}+6tx+3t$$

Lahutage mõlemast poolest 3tx^{2}.

$$3x^{2}t+3xt^{2}+t^{3}-3tx^{2}=6tx+3t$$

Kombineerige 3x^{2}t ja -3tx^{2}, et leida 0.

$$3xt^{2}+t^{3}=6tx+3t$$

Lahutage mõlemast poolest 6tx.

$$3xt^{2}+t^{3}-6tx=3t$$

Lahutage mõlemast poolest t^{3}.

$$3xt^{2}-6tx=3t-t^{3}$$

Kombineerige kõik liikmed, mis sisaldavad x.

$$\left(3t^{2}-6t\right)x=3t-t^{3}$$

Jagage mõlemad pooled 3t^{2}-6t-ga.

$$\frac{\left(3t^{2}-6t\right)x}{3t^{2}-6t}=\frac{t\left(3-t^{2}\right)}{3t^{2}-6t}$$

3t^{2}-6t-ga jagamine võtab 3t^{2}-6t-ga korrutamise tagasi.

$$x=\frac{t\left(3-t^{2}\right)}{3t^{2}-6t}$$

Jagage t\left(3-t^{2}\right) väärtusega 3t^{2}-6t.

$$x=\frac{3-t^{2}}{3\left(t-2\right)}$$

Lahendage ja leidke t (complex solution)

$$t=\frac{-\sqrt{9x^{2}+24x+12}-3x}{2}$$

$$t=0$$

$$t=\frac{\sqrt{9x^{2}+24x+12}-3x}{2}$$

Lahendage ja leidke t

$$\left\{\begin{matrix}\\t=0\text{, }&\text{unconditionally}\\t=\frac{\sqrt{9x^{2}+24x+12}-3x}{2}\text{; }t=\frac{-\sqrt{9x^{2}+24x+12}-3x}{2}\text{, }&x\geq -\frac{2}{3}\text{ or }x\leq -2\end{matrix}\right,$$