$x^2+10x+9=0x^2+10x+9=0$

Ergänze quadratisch mit $5^{2}5^2$.

$x^2+x+5^2-5^2+9=0x^2+\backslash underset\{\}\{2\backslash cdot5\}x+5^2-5^2+9=0$

Fasse zusammen.

$(x^2+2\cdot 5x+5^2)-16=0\backslash left(x^2+2\backslash cdot5x+5^2\backslash ;\backslash right)-16=0$

Fasse als 1. binomische Formel zusammen.

${(x+5)}^2-16=0\backslash left(x+5\backslash ;\backslash right)^2-16=0$

Addiere auf beiden Seiten 16

${(x+5)}^2=16\backslash left(x+5\backslash ;\backslash right)^2=16$

Ziehe Wurzel auf beiden Seiten

$x-5=\pm 4x-5=\pm 4$

Forme weiter um

$x_1=4-5=-1x\_1=4-5=-1$

$x_2=-4-5=-9x\_2=-4-5=-9$

Lösungsmenge angeben

$L=\{-1;-9\}L=\backslash left\backslash \{-1\backslash ;;\backslash ;-9\backslash right\backslash \}$

Quadratische Ergänzung

Artikel zum Thema

$\mathrm{0,5}{x}^2-\mathrm{1,5}x-14=00\{,\}5x^2-1\{,\}5x-14=0$

$\mathrm{0,5}0\{,\}5$ ausklammern

$\mathrm{0,5}(x^2-3x)-14=00\{,\}5\backslash left(x^2-3x\backslash right)-14=0$

Quadratische ergänzen

$\mathrm{0,5}(x^2-3x+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2)-14=00\{,\}5\backslash left(x^2-3x+\backslash left(\backslash frac32\backslash right)^2-\backslash left(\backslash frac32\backslash right)^2\backslash right)-14=0$

Zur 2. binomischen Formel zusammenfassen

$\mathrm{0,5}({(x-\frac{3}{2})}^2-\frac{9}{4})-14=00\{,\}5\backslash left(\backslash left(x-\backslash frac32\backslash right)^2-\backslash frac94\backslash right)-14=0$

Ausmultiplizieren

$\mathrm{0,5}{(x-\frac{3}{2})}^2-\frac{9}{8}-14=00\{,\}5\backslash left(x-\backslash frac32\backslash right)^2-\backslash frac98-14=0$

Gleichung umformen

$\begin{array}{c}\mid +14\\ \mid +\frac{9}{8}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash left|+14\backslash right.\backslash \backslash \backslash left|+\backslash frac98\backslash right.\backslash end\{array\}$

$\mid +\frac{3}{2}\backslash left|+\backslash frac32\backslash right.$

Lösungsmenge angeben

$L=\{-4;7\}L=\backslash left\backslash \{-4;\backslash ;7\backslash right\backslash \}$

$-\frac{1}{2}{x}^2+7x+\mathrm{7,5}=0-\backslash frac12x^2+7x+7\{,\}5=0$

Klammere $-\frac{1}{2}-\backslash frac12$ vor den x-Termen aus

$-\frac{1}{2}(x^2-14x)+\mathrm{7,5}=0-\backslash frac12\backslash left(x^2-14x\backslash right)+7\{,\}5=0$

Ergänze quadratisch mit $7^{2}7^2$

$-\frac{1}{2}(x^2-2\cdot 7x+7^2-7^2)+\mathrm{7,5}=0-\backslash frac12\backslash left(x^2-2\backslash cdot7x+7^2\backslash ;-7^2\backslash right)+7\{,\}5=0$

.Fasse als 2. binomische Formel zusammen.

$-\frac{1}{2}({(x-7)}^2-49)+\mathrm{7,5}=0-\backslash frac12\backslash left(\backslash left(x-7\backslash right)^2-49\backslash right)+7\{,\}5=0$

Multipliziere Klammer aus

$-\frac{1}{2}\left({(x-7)}^2\right)+\mathrm{24,5}+\mathrm{7,5}=0-\backslash frac12\backslash left(\backslash left(x-7\backslash right)^2\backslash right)+24\{,\}5+7\{,\}5=0$

Fasse zusammen.

$-\frac{1}{2}{(x-7)}^2+32=0-\backslash frac12\backslash left(x-7\backslash right)^2+32=0$

Subtrahiere auf beiden Seiten 32

$-\frac{1}{2}{(x-7)}^2=-32-\backslash frac12\backslash left(x-7\backslash ;\backslash right)^2=-32$

Multipliziere beide Seiten mit -2

${(x-7)}^2=64\backslash left(x-7\backslash ;\backslash right)^2=64$

Ziehe Wurzel auf beiden Seiten

$x-7=\pm 8x-7=\backslash pm8$

Forme weiter um

$x_1=8+7=15x\_1=8+7=15$                   $x_2=-8+7=-1x\_2=-8+7=-1$

Gib die Lösungsmenge an

$L=\{-1;15\}L=\backslash left\backslash \{-1\backslash ;;\backslash ;15\backslash right\backslash \}$

$2x^2+2x-\frac{8}{9}=02x^2+2x-\backslash frac89=0$

Klammere 2 vor den x-Termen aus

$2(x^2+x)-\frac{8}{9}=02\backslash left(x^2+x\backslash right)-\backslash frac89=0$

Ergänze quadratisch mit ${\left(\frac{1}{2}\right)}^2\backslash left(\backslash frac12\backslash right)^2$

$2(x^2+2\cdot \frac{1}{2}x+{\left(\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2)-\frac{8}{9}=02\backslash left(x^2+2\backslash cdot\backslash frac12x+\backslash left(\backslash frac12\backslash right)^2-\backslash left(\backslash frac12\backslash right)^2\backslash right)-\backslash frac89=0$

Fasse zu 1. binomische Formel zusammen.

$2({(x+\frac{1}{2})}^2-\frac{1}{4})-\frac{8}{9}=02\backslash left(\backslash left(x+\backslash frac12\backslash right)^2-\backslash frac14\backslash right)-\backslash frac89=0$

Multipliziere Klammer aus.

$2{(x+\frac{1}{2})}^2-\frac{1}{2}-\frac{16}{18}=02\backslash left(x+\backslash frac12\backslash right)^2-\backslash frac12-\backslash frac\{16\}\{18\}=0$

Erweitere Brüche auf gemeinsamen Nenner

$2{(x+\frac{1}{2})}^2-\frac{9}{18}-\frac{16}{18}=02\backslash left(x+\backslash frac12\backslash right)^2-\backslash frac9\{18\}-\backslash frac\{16\}\{18\}=0$

Fasse zusammen

$2{(x+\frac{1}{2})}^2-\frac{25}{18}=02\backslash left(x+\backslash frac12\backslash right)^2-\backslash frac\{25\}\{18\}=0$

Addiere auf beiden Seiten $\frac{25}{18}\backslash frac\{25\}\{18\}$

$2{(x+\frac{1}{2})}^2=\frac{25}{18}2\backslash left(x+\backslash frac12\backslash right)^2=\backslash frac\{25\}\{18\}$

Multipliziere beide Seiten mit $\frac{1}{2}\backslash frac12$

${(x+\frac{1}{2})}^2=\frac{25}{36}\backslash left(x+\backslash frac12\backslash right)^2=\backslash frac\{25\}\{36\}$

Ziehe Wurzel auf beiden Seiten

$x+\frac{1}{2}=\pm \frac{5}{6}x+\backslash frac12=\backslash pm\backslash frac56$

Forme weiter um

$x_1=\frac{5}{6}-\frac{1}{2}=\frac{5}{6}-\frac{3}{6}=\frac{2}{6}=\frac{1}{3}x\_1=\backslash frac56-\backslash frac12=\backslash frac56-\backslash frac36=\backslash frac26\backslash ;=\backslash frac13$                   $x_2=-\frac{5}{6}-\frac{1}{2}=-\frac{5}{6}-\frac{3}{6}=-\frac{8}{6}=-\frac{4}{3}x\_2=-\backslash frac56-\backslash frac12=-\backslash frac56-\backslash frac36=-\backslash frac86\backslash ;=-\backslash frac43$

Lösungsmenge angeben

$L=\{-\frac{4}{3};\frac{1}{3}\}L=\backslash left\backslash \{-\backslash frac43\backslash ;;\backslash ;\backslash frac13\backslash right\backslash \}$

Quadratische Ergänzung

Artikel zum Thema

$2x^2=x+12x^2=x+1$

Alles auf eine Seite bringen

$2x^2-x-1=02x^2-x-1=0$

2 ausklammern.

$2(x^2-\frac{1}{2}x)-1=02\backslash left(x^2-\backslash frac12x\backslash right)-1=0$

Quadratisch ergänzen

$2(x^2-\frac{1}{2}x+{\left(\frac{1}{4}\right)}^2-{\left(\frac{1}{4}\right)}^2)-1=02\backslash left(x^2-\backslash frac12x+\backslash left(\backslash frac14\backslash right)^2-\backslash left(\backslash frac14\backslash right)^2\backslash right)-1=0$

Zur 2. binomischen Formel zusammenfassen.

$2({(x-\frac{1}{4})}^2-\frac{1}{16})-1=02\backslash left(\backslash left(x-\backslash frac14\backslash right)^2-\backslash frac1\{16\}\backslash right)-1=0$

Klammer ausmultiplizieren

$2{(x-\frac{1}{4})}^2-\frac{1}{8}-1=02\backslash left(x-\backslash frac14\backslash right)^2-\backslash frac18-1=0$

Gleichung nach x auflösen

Lösungsmenge angeben

$L=\{-\frac{1}{2};1\}L=\backslash left\backslash \{-\backslash frac12;\backslash ;1\backslash right\backslash \}$