Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(\frac{11}{4}x=-\frac{1}{4}x^2-\frac{5}{2}\)
- \(-\frac{7}{5}x=-\frac{1}{10}x^2-\frac{24}{5}\)
- \(3x^2-(9x+8)=x(x-24)\)
- \(x(x+49)=45(x+1)\)
- \(x(9x-7)=-(x+1)\)
- \(-(15-24x)=-x^2-(40-18x)\)
- \(x(16x+63)=7(x-7)\)
- \(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}=0\)
- \(x(3x+55)=48(x+1)\)
- \(-(4-54x)=-4x^2-(125-10x)\)
- \((-x+3)(-2x-5)-x(-2x-38)=-64\)
- \((-4x-1)(x-1)-x(-28x-2)=7\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\frac{11}{4}x=-\frac{1}{4}x^2-\frac{5}{2} \\
\Leftrightarrow \frac{1}{4}x^2+\frac{11}{4}x+\frac{5}{2}=0 \\
\Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{11}{4}x+\frac{5}{2}\right)=0 \color{red}{.4} \\
\Leftrightarrow x^2+11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+10=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (11)^2-4.1.10 & &\\
& = 121-40 & & \\
& = 81 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-11-\sqrt81}{2.1} & & = \frac{-11+\sqrt81}{2.1} \\
& = \frac{-20}{2} & & = \frac{-2}{2} \\
& = -10 & & = -1 \\ \\ V &= \Big\{ -10 ; -1 \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{7}{5}x=-\frac{1}{10}x^2-\frac{24}{5} \\
\Leftrightarrow \frac{1}{10}x^2-\frac{7}{5}x+\frac{24}{5}=0 \\
\Leftrightarrow \color{red}{10.} \left(\frac{1}{10}x^2-\frac{7}{5}x+\frac{24}{5}\right)=0 \color{red}{.10} \\
\Leftrightarrow x^2-14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-14)^2-4.1.48 & &\\
& = 196-192 & & \\
& = 4 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-14)-\sqrt4}{2.1} & & = \frac{-(-14)+\sqrt4}{2.1} \\
& = \frac{12}{2} & & = \frac{16}{2} \\
& = 6 & & = 8 \\ \\ V &= \Big\{ 6 ; 8 \Big\} & &\end{align} \\ -----------------\)
- \(3x^2-(9x+8)=x(x-24) \\
\Leftrightarrow 3x^2-9x-8=x^2-24x \\
\Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.2.(-8) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\
& = \frac{-32}{4} & & = \frac{2}{4} \\
& = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(x(x+49)=45(x+1) \\
\Leftrightarrow x^2+49x=45x+45 \\
\Leftrightarrow x^2+4x-45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-45=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (4)^2-4.1.(-45) & &\\
& = 16+180 & & \\
& = 196 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-4-\sqrt196}{2.1} & & = \frac{-4+\sqrt196}{2.1} \\
& = \frac{-18}{2} & & = \frac{10}{2} \\
& = -9 & & = 5 \\ \\ V &= \Big\{ -9 ; 5 \Big\} & &\end{align} \\ -----------------\)
- \(x(9x-7)=-(x+1) \\
\Leftrightarrow 9x^2-7x=-x-1 \\
\Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-6)^2-4.9.1 & &\\
& = 36-36 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-6)}{2.9} & & \\
& = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
- \(-(15-24x)=-x^2-(40-18x) \\
\Leftrightarrow -15+24x=-x^2-40+18x \\
\Leftrightarrow x^2+6x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (6)^2-4.1.25 & &\\
& = 36-100 & & \\
& = -64 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(x(16x+63)=7(x-7) \\
\Leftrightarrow 16x^2+63x=7x-49 \\
\Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (56)^2-4.16.49 & &\\
& = 3136-3136 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-56}{2.16} & & \\
& = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}=0\\
\Leftrightarrow \color{red}{60.} \left(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}\right)=0 \color{red}{.60} \\
\Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (48)^2-4.4.144 & &\\
& = 2304-2304 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-48}{2.4} & & \\
& = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
- \(x(3x+55)=48(x+1) \\
\Leftrightarrow 3x^2+55x=48x+48 \\
\Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.3.(-48) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\
& = \frac{-32}{6} & & = \frac{18}{6} \\
& = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(-(4-54x)=-4x^2-(125-10x) \\
\Leftrightarrow -4+54x=-4x^2-125+10x \\
\Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (44)^2-4.4.121 & &\\
& = 1936-1936 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-44}{2.4} & & \\
& = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
- \((-x+3)(-2x-5)-x(-2x-38)=-64\\
\Leftrightarrow 2x^2+5x-6x-15 +2x^2+38x+64=0 \\
\Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (28)^2-4.4.49 & &\\
& = 784-784 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-28}{2.4} & & \\
& = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
- \((-4x-1)(x-1)-x(-28x-2)=7\\
\Leftrightarrow -4x^2+4x-x+1 +28x^2+2x-7=0 \\
\Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.24.(-6) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\
& = \frac{-32}{48} & & = \frac{18}{48} \\
& = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)