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

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

Solvi għal x (complex solution)

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

Passi biex Issolvi l-Ekwazzjoni Lineari

Uża teorema binomjali \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} biex tespandi \left(x+t\right)^{3}.

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

Ikkombina x^{3} u -x^{3} biex tikseb 0.

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

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(x+1\right)^{2}.

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

Uża l-propjetà distributtiva biex timmultiplika 3t b'x^{2}+2x+1.

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

Naqqas 3tx^{2} miż-żewġ naħat.

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

Ikkombina 3x^{2}t u -3tx^{2} biex tikseb 0.

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

Naqqas 6tx miż-żewġ naħat.

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

Naqqas t^{3} miż-żewġ naħat.

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

Ikkombina t-termini kollha li fihom x.

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

Iddividi ż-żewġ naħat b'3t^{2}-6t.

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

Meta tiddividi b'3t^{2}-6t titneħħa l-multiplikazzjoni b'3t^{2}-6t.

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

Iddividi t\left(3-t^{2}\right) b'3t^{2}-6t.

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

Solvi għal x

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

Passi biex Issolvi l-Ekwazzjoni Lineari

Uża teorema binomjali \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} biex tespandi \left(x+t\right)^{3}.

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

Ikkombina x^{3} u -x^{3} biex tikseb 0.

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

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(x+1\right)^{2}.

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

Uża l-propjetà distributtiva biex timmultiplika 3t b'x^{2}+2x+1.

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

Naqqas 3tx^{2} miż-żewġ naħat.

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

Ikkombina 3x^{2}t u -3tx^{2} biex tikseb 0.

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

Naqqas 6tx miż-żewġ naħat.

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

Naqqas t^{3} miż-żewġ naħat.

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

Ikkombina t-termini kollha li fihom x.

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

Iddividi ż-żewġ naħat b'3t^{2}-6t.

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

Meta tiddividi b'3t^{2}-6t titneħħa l-multiplikazzjoni b'3t^{2}-6t.

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

Iddividi t\left(3-t^{2}\right) b'3t^{2}-6t.

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

Solvi għal t (complex solution)

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

$$t=0$$

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

Solvi għal t

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