Issolvi 2*(3x+4)-2*(5x+2)=3:(2x+2)+4*(4x+10)

$$2 \cdot ( 3 x + 4 ) - 2 \cdot ( 5 x + 2 ) = 3 : ( 2 x + 2 ) + 4 \cdot ( 4 x + 10 )$$

Solvi għal x

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5}\approx -1.108452405$$

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5}\approx -1.691547595$$

Passi għall-Użu tal-Formula Kwadratika

Il-varjabbli x ma jistax ikun ugwali għal -1 billi d-diviżjoni b'żero mhux iddefinit. Immultiplika ż-żewġ naħat tal-ekwazzjoni b'2\left(x+1\right).

$$2\left(3x+4\right)\times 2\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Immultiplika 2 u 2 biex tikseb 4.

$$4\left(3x+4\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 4 b'3x+4.

$$\left(12x+16\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 12x+16 b'x+1 u kkombina termini simili.

$$12x^{2}+28x+16-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Immultiplika -2 u 2 biex tikseb -4.

$$12x^{2}+28x+16-4\left(5x+2\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika -4 b'5x+2.

$$12x^{2}+28x+16+\left(-20x-8\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika -20x-8 b'x+1 u kkombina termini simili.

$$12x^{2}+28x+16-20x^{2}-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Ikkombina 12x^{2} u -20x^{2} biex tikseb -8x^{2}.

$$-8x^{2}+28x+16-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Ikkombina 28x u -28x biex tikseb 0.

$$-8x^{2}+16-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Naqqas 8 minn 16 biex tikseb 8.

$$-8x^{2}+8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Immultiplika 4 u 2 biex tikseb 8.

$$-8x^{2}+8=3+8\left(4x+10\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 8 b'4x+10.

$$-8x^{2}+8=3+\left(32x+80\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 32x+80 b'x+1 u kkombina termini simili.

$$-8x^{2}+8=3+32x^{2}+112x+80$$

Żid 3 u 80 biex tikseb 83.

$$-8x^{2}+8=83+32x^{2}+112x$$

Naqqas 83 miż-żewġ naħat.

$$-8x^{2}+8-83=32x^{2}+112x$$

Naqqas 83 minn 8 biex tikseb -75.

$$-8x^{2}-75=32x^{2}+112x$$

Naqqas 32x^{2} miż-żewġ naħat.

$$-8x^{2}-75-32x^{2}=112x$$

Ikkombina -8x^{2} u -32x^{2} biex tikseb -40x^{2}.

$$-40x^{2}-75=112x$$

Naqqas 112x miż-żewġ naħat.

$$-40x^{2}-75-112x=0$$

L-ekwazzjonijiet kollha tal-formola ax^{2}+bx+c=0 jistgħu jiġu solvuti permezz tal-formula kwadratika: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Il-formula kwadratika tagħti żewġ soluzzjonijiet, waħda meta ± hija addizzjoni u waħda meta hija tnaqqis.

$$-40x^{2}-112x-75=0$$

Din l-ekwazzjoni hija fil-forma standard: ax^{2}+bx+c=0. Issostitwixxi -40 għal a, -112 għal b, u -75 għal c fil-formula kwadratika, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\left(-40\right)\left(-75\right)}}{2\left(-40\right)}$$

Ikkwadra -112.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\left(-40\right)\left(-75\right)}}{2\left(-40\right)}$$

Immultiplika -4 b'-40.

$$x=\frac{-\left(-112\right)±\sqrt{12544+160\left(-75\right)}}{2\left(-40\right)}$$

Immultiplika 160 b'-75.

$$x=\frac{-\left(-112\right)±\sqrt{12544-12000}}{2\left(-40\right)}$$

Żid 12544 ma' -12000.

$$x=\frac{-\left(-112\right)±\sqrt{544}}{2\left(-40\right)}$$

Ħu l-għerq kwadrat ta' 544.

$$x=\frac{-\left(-112\right)±4\sqrt{34}}{2\left(-40\right)}$$

L-oppost ta' -112 huwa 112.

$$x=\frac{112±4\sqrt{34}}{2\left(-40\right)}$$

Immultiplika 2 b'-40.

$$x=\frac{112±4\sqrt{34}}{-80}$$

Issa solvi l-ekwazzjoni x=\frac{112±4\sqrt{34}}{-80} fejn ± hija plus. Żid 112 ma' 4\sqrt{34}.

$$x=\frac{4\sqrt{34}+112}{-80}$$

Iddividi 112+4\sqrt{34} b'-80.

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Issa solvi l-ekwazzjoni x=\frac{112±4\sqrt{34}}{-80} fejn ± hija minus. Naqqas 4\sqrt{34} minn 112.

$$x=\frac{112-4\sqrt{34}}{-80}$$

Iddividi 112-4\sqrt{34} b'-80.

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5}$$

L-ekwazzjoni issa solvuta.

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5} x=\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Passi biex Tikkompleta l-Kwadrat

Il-varjabbli x ma jistax ikun ugwali għal -1 billi d-diviżjoni b'żero mhux iddefinit. Immultiplika ż-żewġ naħat tal-ekwazzjoni b'2\left(x+1\right).

$$2\left(3x+4\right)\times 2\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Immultiplika 2 u 2 biex tikseb 4.

$$4\left(3x+4\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 4 b'3x+4.

$$\left(12x+16\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 12x+16 b'x+1 u kkombina termini simili.

$$12x^{2}+28x+16-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Immultiplika -2 u 2 biex tikseb -4.

$$12x^{2}+28x+16-4\left(5x+2\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika -4 b'5x+2.

$$12x^{2}+28x+16+\left(-20x-8\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika -20x-8 b'x+1 u kkombina termini simili.

$$12x^{2}+28x+16-20x^{2}-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Ikkombina 12x^{2} u -20x^{2} biex tikseb -8x^{2}.

$$-8x^{2}+28x+16-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Ikkombina 28x u -28x biex tikseb 0.

$$-8x^{2}+16-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Naqqas 8 minn 16 biex tikseb 8.

$$-8x^{2}+8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

Immultiplika 4 u 2 biex tikseb 8.

$$-8x^{2}+8=3+8\left(4x+10\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 8 b'4x+10.

$$-8x^{2}+8=3+\left(32x+80\right)\left(x+1\right)$$

Uża l-propjetà distributtiva biex timmultiplika 32x+80 b'x+1 u kkombina termini simili.

$$-8x^{2}+8=3+32x^{2}+112x+80$$

Żid 3 u 80 biex tikseb 83.

$$-8x^{2}+8=83+32x^{2}+112x$$

Naqqas 32x^{2} miż-żewġ naħat.

$$-8x^{2}+8-32x^{2}=83+112x$$

Ikkombina -8x^{2} u -32x^{2} biex tikseb -40x^{2}.

$$-40x^{2}+8=83+112x$$

Naqqas 112x miż-żewġ naħat.

$$-40x^{2}+8-112x=83$$

Naqqas 8 miż-żewġ naħat.

$$-40x^{2}-112x=83-8$$

Naqqas 8 minn 83 biex tikseb 75.

$$-40x^{2}-112x=75$$

Iddividi ż-żewġ naħat b'-40.

$$\frac{-40x^{2}-112x}{-40}=\frac{75}{-40}$$

Meta tiddividi b'-40 titneħħa l-multiplikazzjoni b'-40.

$$x^{2}+\left(-\frac{112}{-40}\right)x=\frac{75}{-40}$$

Naqqas il-frazzjoni \frac{-112}{-40} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 8.

$$x^{2}+\frac{14}{5}x=\frac{75}{-40}$$

Naqqas il-frazzjoni \frac{75}{-40} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 5.

$$x^{2}+\frac{14}{5}x=-\frac{15}{8}$$

Iddividi \frac{14}{5}, il-koeffiċjent tat-terminu x, b'2 biex tikseb \frac{7}{5}. Imbagħad żid il-kwadru ta' \frac{7}{5} maż-żewġ naħat tal-ekwazzjoni. Dan il-pass jagħmel in-naħa tax-xellug tal-ekwazzjoni kwadru perfett.

$$x^{2}+\frac{14}{5}x+\left(\frac{7}{5}\right)^{2}=-\frac{15}{8}+\left(\frac{7}{5}\right)^{2}$$

Ikkwadra \frac{7}{5} billi tikkwadra kemm in-numeratur u d-denominatur tal-frazzjoni.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=-\frac{15}{8}+\frac{49}{25}$$

Żid -\frac{15}{8} ma' \frac{49}{25} biex issib id-denominatur komuni u żżid in-numeraturi. Imbagħad naqqas il-frazzjoni għat-termini l-aktar baxxi jekk possibbli.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{17}{200}$$

Fattur x^{2}+\frac{14}{5}x+\frac{49}{25}. B'mod ġenerali, meta x^{2}+bx+c huwa kwadru perfett, dejjem jista' jiġu fatturati bħala \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{5}\right)^{2}=\frac{17}{200}$$

Ħu l-għerq kwadrat taż-żewġ naħat tal-ekwazzjoni.

$$\sqrt{\left(x+\frac{7}{5}\right)^{2}}=\sqrt{\frac{17}{200}}$$

Issimplifika.

$$x+\frac{7}{5}=\frac{\sqrt{34}}{20} x+\frac{7}{5}=-\frac{\sqrt{34}}{20}$$

Naqqas \frac{7}{5} miż-żewġ naħat tal-ekwazzjoni.

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5} x=-\frac{\sqrt{34}}{20}-\frac{7}{5}$$