Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(20x^2-(9x-48)=17x(x-2)\)
- \(3x^2-(20x-33)=2x(x-17)\)
- \((-2x+3)(-4x+3)-x(7x+12)=-11\)
- \(x(x+55)=50(x+1)\)
- \(\frac{1}{5}x=-\frac{1}{100}x^2-1\)
- \((-4x-2)(-3x+4)-x(-4x-120)=-152\)
- \(\frac{7}{5}x=-\frac{1}{5}x^2+6\)
- \(-(3-9x)=-x^2-(-11-14x)\)
- \(7x^2-(17x+48)=4x(x-6)\)
- \(2x^2-(18x+96)=x(x-14)\)
- \((-5x-1)(2x+3)-x(-19x-24)=-39\)
- \(\frac{1}{11}x^2-x+\frac{11}{4}=0\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(20x^2-(9x-48)=17x(x-2) \\
\Leftrightarrow 20x^2-9x+48=17x^2-34x \\
\Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.3.48 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\
& = \frac{-32}{6} & & = \frac{-18}{6} \\
& = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
- \(3x^2-(20x-33)=2x(x-17) \\
\Leftrightarrow 3x^2-20x+33=2x^2-34x \\
\Leftrightarrow x^2+14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+33=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (14)^2-4.1.33 & &\\
& = 196-132 & & \\
& = 64 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-14-\sqrt64}{2.1} & & = \frac{-14+\sqrt64}{2.1} \\
& = \frac{-22}{2} & & = \frac{-6}{2} \\
& = -11 & & = -3 \\ \\ V &= \Big\{ -11 ; -3 \Big\} & &\end{align} \\ -----------------\)
- \((-2x+3)(-4x+3)-x(7x+12)=-11\\
\Leftrightarrow 8x^2-6x-12x+9 -7x^2-12x+11=0 \\
\Leftrightarrow x^2-9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+20=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-9)^2-4.1.20 & &\\
& = 81-80 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-9)-\sqrt1}{2.1} & & = \frac{-(-9)+\sqrt1}{2.1} \\
& = \frac{8}{2} & & = \frac{10}{2} \\
& = 4 & & = 5 \\ \\ V &= \Big\{ 4 ; 5 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+55)=50(x+1) \\
\Leftrightarrow x^2+55x=50x+50 \\
\Leftrightarrow x^2+5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-50=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.1.(-50) & &\\
& = 25+200 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt225}{2.1} & & = \frac{-5+\sqrt225}{2.1} \\
& = \frac{-20}{2} & & = \frac{10}{2} \\
& = -10 & & = 5 \\ \\ V &= \Big\{ -10 ; 5 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{5}x=-\frac{1}{100}x^2-1 \\
\Leftrightarrow \frac{1}{100}x^2+\frac{1}{5}x+1=0 \\
\Leftrightarrow \color{red}{100.} \left(\frac{1}{100}x^2+\frac{1}{5}x+1\right)=0 \color{red}{.100} \\
\Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (20)^2-4.1.100 & &\\
& = 400-400 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-20}{2.1} & & \\
& = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
- \((-4x-2)(-3x+4)-x(-4x-120)=-152\\
\Leftrightarrow 12x^2-16x+6x-8 +4x^2+120x+152=0 \\
\Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (96)^2-4.16.144 & &\\
& = 9216-9216 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-96}{2.16} & & \\
& = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{7}{5}x=-\frac{1}{5}x^2+6 \\
\Leftrightarrow \frac{1}{5}x^2+\frac{7}{5}x-6=0 \\
\Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{7}{5}x-6\right)=0 \color{red}{.5} \\
\Leftrightarrow x^2+7x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-30=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.1.(-30) & &\\
& = 49+120 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt169}{2.1} & & = \frac{-7+\sqrt169}{2.1} \\
& = \frac{-20}{2} & & = \frac{6}{2} \\
& = -10 & & = 3 \\ \\ V &= \Big\{ -10 ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(-(3-9x)=-x^2-(-11-14x) \\
\Leftrightarrow -3+9x=-x^2+11+14x \\
\Leftrightarrow x^2-5x-14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-14=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-5)^2-4.1.(-14) & &\\
& = 25+56 & & \\
& = 81 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-5)-\sqrt81}{2.1} & & = \frac{-(-5)+\sqrt81}{2.1} \\
& = \frac{-4}{2} & & = \frac{14}{2} \\
& = -2 & & = 7 \\ \\ V &= \Big\{ -2 ; 7 \Big\} & &\end{align} \\ -----------------\)
- \(7x^2-(17x+48)=4x(x-6) \\
\Leftrightarrow 7x^2-17x-48=4x^2-24x \\
\Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.3.(-48) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\
& = \frac{-32}{6} & & = \frac{18}{6} \\
& = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(18x+96)=x(x-14) \\
\Leftrightarrow 2x^2-18x-96=x^2-14x \\
\Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-4)^2-4.1.(-96) & &\\
& = 16+384 & & \\
& = 400 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\
& = \frac{-16}{2} & & = \frac{24}{2} \\
& = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \((-5x-1)(2x+3)-x(-19x-24)=-39\\
\Leftrightarrow -10x^2-15x-2x-3 +19x^2+24x+39=0 \\
\Leftrightarrow 9x^2+6x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (6)^2-4.9.36 & &\\
& = 36-1296 & & \\
& = -1260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(\frac{1}{11}x^2-x+\frac{11}{4}=0\\
\Leftrightarrow \color{red}{44.} \left(\frac{1}{11}x^2-x+\frac{11}{4}\right)=0 \color{red}{.44} \\
\Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-44)^2-4.4.121 & &\\
& = 1936-1936 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-44)}{2.4} & & \\
& = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)