Issolvi 5nx^2-5x-nx-1/x^2-1=2n+1

$$\frac { 5 n x ^ { 2 } - 5 x - n x - 1 } { x ^ { 2 } - 1 } = 2 n + 1$$

Solvi għal n (complex solution)

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

$$x\neq \frac{1+\sqrt{23}i}{6}\text{ and }x\neq \frac{-\sqrt{23}i+1}{6}\text{ and }x\neq -1\text{ and }x\neq 1$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Immultiplika ż-żewġ naħat tal-ekwazzjoni b'\left(x-1\right)\left(x+1\right).

$$5nx^{2}-5x-nx-1=2n\left(x-1\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 2n b'x-1.

$$5nx^{2}-5x-nx-1=\left(2nx-2n\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 2nx-2n b'x+1 u kkombina termini simili.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+\left(x-1\right)\left(x+1\right)$$

Ikkunsidra li \left(x-1\right)\left(x+1\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Ikkwadra 1.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+x^{2}-1$$

Naqqas 2nx^{2} miż-żewġ naħat.

$$5nx^{2}-5x-nx-1-2nx^{2}=-2n+x^{2}-1$$

Ikkombina 5nx^{2} u -2nx^{2} biex tikseb 3nx^{2}.

$$3nx^{2}-5x-nx-1=-2n+x^{2}-1$$

Żid 2n maż-żewġ naħat.

$$3nx^{2}-5x-nx-1+2n=x^{2}-1$$

Żid 5x maż-żewġ naħat.

$$3nx^{2}-nx-1+2n=x^{2}-1+5x$$

Żid 1 maż-żewġ naħat.

$$3nx^{2}-nx+2n=x^{2}-1+5x+1$$

Żid -1 u 1 biex tikseb 0.

$$3nx^{2}-nx+2n=x^{2}+5x$$

Ikkombina t-termini kollha li fihom n.

$$\left(3x^{2}-x+2\right)n=x^{2}+5x$$

Iddividi ż-żewġ naħat b'3x^{2}-x+2.

$$\frac{\left(3x^{2}-x+2\right)n}{3x^{2}-x+2}=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Meta tiddividi b'3x^{2}-x+2 titneħħa l-multiplikazzjoni b'3x^{2}-x+2.

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Solvi għal n

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

$$|x|\neq 1$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Immultiplika ż-żewġ naħat tal-ekwazzjoni b'\left(x-1\right)\left(x+1\right).

$$5nx^{2}-5x-nx-1=2n\left(x-1\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 2n b'x-1.

$$5nx^{2}-5x-nx-1=\left(2nx-2n\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 2nx-2n b'x+1 u kkombina termini simili.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+\left(x-1\right)\left(x+1\right)$$

Ikkunsidra li \left(x-1\right)\left(x+1\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Ikkwadra 1.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+x^{2}-1$$

Naqqas 2nx^{2} miż-żewġ naħat.

$$5nx^{2}-5x-nx-1-2nx^{2}=-2n+x^{2}-1$$

Ikkombina 5nx^{2} u -2nx^{2} biex tikseb 3nx^{2}.

$$3nx^{2}-5x-nx-1=-2n+x^{2}-1$$

Żid 2n maż-żewġ naħat.

$$3nx^{2}-5x-nx-1+2n=x^{2}-1$$

Żid 5x maż-żewġ naħat.

$$3nx^{2}-nx-1+2n=x^{2}-1+5x$$

Żid 1 maż-żewġ naħat.

$$3nx^{2}-nx+2n=x^{2}-1+5x+1$$

Żid -1 u 1 biex tikseb 0.

$$3nx^{2}-nx+2n=x^{2}+5x$$

Ikkombina t-termini kollha li fihom n.

$$\left(3x^{2}-x+2\right)n=x^{2}+5x$$

Iddividi ż-żewġ naħat b'3x^{2}-x+2.

$$\frac{\left(3x^{2}-x+2\right)n}{3x^{2}-x+2}=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Meta tiddividi b'3x^{2}-x+2 titneħħa l-multiplikazzjoni b'3x^{2}-x+2.

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Solvi għal x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq \frac{1}{3}\\x=\frac{\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq -\frac{2}{3}\text{ and }n\neq \frac{1}{3}\text{ and }n\neq \frac{3}{2}\\x=\frac{1}{8}\text{, }&n=\frac{1}{3}\end{matrix}\right.$$

Solvi għal x

$$\left\{\begin{matrix}x=\frac{-\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq \frac{1}{3}\text{ and }n\geq \frac{9-4\sqrt{41}}{23}\text{ and }n\leq \frac{4\sqrt{41}+9}{23}\\x=\frac{\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq \frac{3}{2}\text{ and }n\neq \frac{1}{3}\text{ and }n\geq \frac{9-4\sqrt{41}}{23}\text{ and }n\leq \frac{4\sqrt{41}+9}{23}\text{ and }n\neq -\frac{2}{3}\\x=\frac{1}{8}\text{, }&n=\frac{1}{3}\end{matrix}\right.$$