Issolvi (2y+3)^2+y^2=4

$$( 2 y + 3 ) ^ { 2 } + y ^ { 2 } = 4$$

Solvi għal y

$$y=\frac{\sqrt{11}-6}{5}\approx -0.536675042$$

$$y=\frac{-\sqrt{11}-6}{5}\approx -1.863324958$$

Passi għall-Użu tal-Formula Kwadratika

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

$$4y^{2}+12y+9+y^{2}=4$$

Ikkombina 4y^{2} u y^{2} biex tikseb 5y^{2}.

$$5y^{2}+12y+9=4$$

Naqqas 4 miż-żewġ naħat.

$$5y^{2}+12y+9-4=0$$

Naqqas 4 minn 9 biex tikseb 5.

$$5y^{2}+12y+5=0$$

Din l-ekwazzjoni hija fil-forma standard: ax^{2}+bx+c=0. Issostitwixxi 5 għal a, 12 għal b, u 5 għal c fil-formula kwadratika, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-12±\sqrt{12^{2}-4\times 5\times 5}}{2\times 5}$$

Ikkwadra 12.

$$y=\frac{-12±\sqrt{144-4\times 5\times 5}}{2\times 5}$$

Immultiplika -4 b'5.

$$y=\frac{-12±\sqrt{144-20\times 5}}{2\times 5}$$

Immultiplika -20 b'5.

$$y=\frac{-12±\sqrt{144-100}}{2\times 5}$$

Żid 144 ma' -100.

$$y=\frac{-12±\sqrt{44}}{2\times 5}$$

Ħu l-għerq kwadrat ta' 44.

$$y=\frac{-12±2\sqrt{11}}{2\times 5}$$

Immultiplika 2 b'5.

$$y=\frac{-12±2\sqrt{11}}{10}$$

Issa solvi l-ekwazzjoni y=\frac{-12±2\sqrt{11}}{10} fejn ± hija plus. Żid -12 ma' 2\sqrt{11}.

$$y=\frac{2\sqrt{11}-12}{10}$$

Iddividi -12+2\sqrt{11} b'10.

$$y=\frac{\sqrt{11}-6}{5}$$

Issa solvi l-ekwazzjoni y=\frac{-12±2\sqrt{11}}{10} fejn ± hija minus. Naqqas 2\sqrt{11} minn -12.

$$y=\frac{-2\sqrt{11}-12}{10}$$

Iddividi -12-2\sqrt{11} b'10.

$$y=\frac{-\sqrt{11}-6}{5}$$

L-ekwazzjoni issa solvuta.

$$y=\frac{\sqrt{11}-6}{5} y=\frac{-\sqrt{11}-6}{5}$$

Passi biex Tikkompleta l-Kwadrat

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

$$4y^{2}+12y+9+y^{2}=4$$

Ikkombina 4y^{2} u y^{2} biex tikseb 5y^{2}.

$$5y^{2}+12y+9=4$$

Naqqas 9 miż-żewġ naħat.

$$5y^{2}+12y=4-9$$

Naqqas 9 minn 4 biex tikseb -5.

$$5y^{2}+12y=-5$$

Iddividi ż-żewġ naħat b'5.

$$\frac{5y^{2}+12y}{5}=-\frac{5}{5}$$

Meta tiddividi b'5 titneħħa l-multiplikazzjoni b'5.

$$y^{2}+\frac{12}{5}y=-\frac{5}{5}$$

Iddividi -5 b'5.

$$y^{2}+\frac{12}{5}y=-1$$

Iddividi \frac{12}{5}, il-koeffiċjent tat-terminu x, b'2 biex tikseb \frac{6}{5}. Imbagħad żid il-kwadru ta' \frac{6}{5} maż-żewġ naħat tal-ekwazzjoni. Dan il-pass jagħmel in-naħa tax-xellug tal-ekwazzjoni kwadru perfett.

$$y^{2}+\frac{12}{5}y+\left(\frac{6}{5}\right)^{2}=-1+\left(\frac{6}{5}\right)^{2}$$

Ikkwadra \frac{6}{5} billi tikkwadra kemm in-numeratur u d-denominatur tal-frazzjoni.

$$y^{2}+\frac{12}{5}y+\frac{36}{25}=-1+\frac{36}{25}$$

Żid -1 ma' \frac{36}{25}.

$$y^{2}+\frac{12}{5}y+\frac{36}{25}=\frac{11}{25}$$

Fattur y^{2}+\frac{12}{5}y+\frac{36}{25}. B'mod ġenerali, meta x^{2}+bx+c huwa kwadru perfett, dejjem jista' jiġu fatturati bħala \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{6}{5}\right)^{2}=\frac{11}{25}$$

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

$$\sqrt{\left(y+\frac{6}{5}\right)^{2}}=\sqrt{\frac{11}{25}}$$

Issimplifika.

$$y+\frac{6}{5}=\frac{\sqrt{11}}{5} y+\frac{6}{5}=-\frac{\sqrt{11}}{5}$$

Naqqas \frac{6}{5} miż-żewġ naħat tal-ekwazzjoni.

$$y=\frac{\sqrt{11}-6}{5} y=\frac{-\sqrt{11}-6}{5}$$