Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(x^2+18x+80=0\)
- \(24x^2+7x-6=0\)
- \(4x^2+10x+25=0\)
- \(4x^2+25x-5=10x-1\)
- \(3x^2+13x+12=0\)
- \(18x^2+15x-9=10x-7\)
- \(4x^2-2x+1=0\)
- \(x^2+x-6=0\)
- \(2x^2-7x+6=-10x+8\)
- \(x^2+2x-63=0\)
- \(x^2+17x+72=0\)
- \(9x^2+18x+7=5x+3\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+80=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (18)^2-4.1.80 & &\\
& = 324-320 & & \\
& = 4 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-18-\sqrt4}{2.1} & & = \frac{-18+\sqrt4}{2.1} \\
& = \frac{-20}{2} & & = \frac{-16}{2} \\
& = -10 & & = -8 \\ \\ V &= \Big\{ -10 ; -8 \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.24.(-6) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\
& = \frac{-32}{48} & & = \frac{18}{48} \\
& = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+10x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (10)^2-4.4.25 & &\\
& = 100-400 & & \\
& = -300 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(4x^2+25x-5=10x-1\\
\Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.4.(-4) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\
& = \frac{-32}{8} & & = \frac{2}{8} \\
& = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.3.12 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\
& = \frac{-18}{6} & & = \frac{-8}{6} \\
& = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
- \(18x^2+15x-9=10x-7\\
\Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.18.(-2) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\
& = \frac{-18}{36} & & = \frac{8}{36} \\
& = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.4.1 & &\\
& = 4-16 & & \\
& = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+x-6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (1)^2-4.1.(-6) & &\\
& = 1+24 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-1-\sqrt25}{2.1} & & = \frac{-1+\sqrt25}{2.1} \\
& = \frac{-6}{2} & & = \frac{4}{2} \\
& = -3 & & = 2 \\ \\ V &= \Big\{ -3 ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-7x+6=-10x+8\\
\Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.2.(-2) & &\\
& = 9+16 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\
& = \frac{-8}{4} & & = \frac{2}{4} \\
& = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-63=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.1.(-63) & &\\
& = 4+252 & & \\
& = 256 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-2-\sqrt256}{2.1} & & = \frac{-2+\sqrt256}{2.1} \\
& = \frac{-18}{2} & & = \frac{14}{2} \\
& = -9 & & = 7 \\ \\ V &= \Big\{ -9 ; 7 \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+72=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.1.72 & &\\
& = 289-288 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt1}{2.1} & & = \frac{-17+\sqrt1}{2.1} \\
& = \frac{-18}{2} & & = \frac{-16}{2} \\
& = -9 & & = -8 \\ \\ V &= \Big\{ -9 ; -8 \Big\} & &\end{align} \\ -----------------\)
- \(9x^2+18x+7=5x+3\\
\Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.9.4 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\
& = \frac{-18}{18} & & = \frac{-8}{18} \\
& = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)