Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(12x^2+32x+14=7x+2\)
- \(4x^2-4x-4=-7x-3\)
- \(x^2-11x+16=-9x+12\)
- \(x^2-9x+37=-11x+12\)
- \(x^2-23x+30=-11x-6\)
- \(16x^2-62x+59=2x-5\)
- \(36x^2+25x+4=0\)
- \(x^2-6x-6=-11x-10\)
- \(2x^2+37x+75=12x+3\)
- \(16x^2+27x-8=12x-7\)
- \(x^2-4x-12=0\)
- \(x^2+2x+26=-9x+8\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(12x^2+32x+14=7x+2\\
\Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.12.12 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\
& = \frac{-32}{24} & & = \frac{-18}{24} \\
& = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(4x^2-4x-4=-7x-3\\
\Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.4.(-1) & &\\
& = 9+16 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\
& = \frac{-8}{8} & & = \frac{2}{8} \\
& = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x^2-11x+16=-9x+12\\
\Leftrightarrow x^2-2x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.1.4 & &\\
& = 4-16 & & \\
& = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(x^2-9x+37=-11x+12\\
\Leftrightarrow x^2+2x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.1.25 & &\\
& = 4-100 & & \\
& = -96 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(x^2-23x+30=-11x-6\\
\Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-12)^2-4.1.36 & &\\
& = 144-144 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-12)}{2.1} & & \\
& = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
- \(16x^2-62x+59=2x-5\\
\Leftrightarrow 16x^2-64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-64x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-64)^2-4.16.64 & &\\
& = 4096-4096 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-64)}{2.16} & & \\
& = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.36.4 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\
& = \frac{-32}{72} & & = \frac{-18}{72} \\
& = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x^2-6x-6=-11x-10\\
\Leftrightarrow x^2+5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.1.4 & &\\
& = 25-16 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt9}{2.1} & & = \frac{-5+\sqrt9}{2.1} \\
& = \frac{-8}{2} & & = \frac{-2}{2} \\
& = -4 & & = -1 \\ \\ V &= \Big\{ -4 ; -1 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2+37x+75=12x+3\\
\Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.2.72 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\
& = \frac{-32}{4} & & = \frac{-18}{4} \\
& = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
- \(16x^2+27x-8=12x-7\\
\Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.16.(-1) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\
& = \frac{-32}{32} & & = \frac{2}{32} \\
& = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-4)^2-4.1.(-12) & &\\
& = 16+48 & & \\
& = 64 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-4)-\sqrt64}{2.1} & & = \frac{-(-4)+\sqrt64}{2.1} \\
& = \frac{-4}{2} & & = \frac{12}{2} \\
& = -2 & & = 6 \\ \\ V &= \Big\{ -2 ; 6 \Big\} & &\end{align} \\ -----------------\)
- \(x^2+2x+26=-9x+8\\
\Leftrightarrow x^2+11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (11)^2-4.1.18 & &\\
& = 121-72 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-11-\sqrt49}{2.1} & & = \frac{-11+\sqrt49}{2.1} \\
& = \frac{-18}{2} & & = \frac{-4}{2} \\
& = -9 & & = -2 \\ \\ V &= \Big\{ -9 ; -2 \Big\} & &\end{align} \\ -----------------\)