Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(2x^2+5x-13=-10x-5\)
2. \(12x^2+5x-3=0\)
3. \(x^2+13x-79=10x-9\)
4. \(x^2+12x+52=-x+10\)
5. \(16x^2+13x+19=-7x+10\)
6. \(16x^2-12x+49=0\)
7. \(x^2+12x+66=-5x+6\)
8. \(9x^2+27x+12=3x-4\)
9. \(x^2-8x+12=0\)
10. \(16x^2-3x-8=-x-9\)
11. \(x^2-7x+15=-12x+11\)
12. \(x^2+13x+78=-6x-10\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(2x^2+5x-13=-10x-5\\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
3. \(x^2+13x-79=10x-9\\ \Leftrightarrow x^2+3x-70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-70) & &\\ & = 9+280 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt289}{2.1} & & = \frac{-3+\sqrt289}{2.1} \\ & = \frac{-20}{2} & & = \frac{14}{2} \\ & = -10 & & = 7 \\ \\ V &= \Big\{ -10 ; 7 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2+12x+52=-x+10\\ \Leftrightarrow x^2+13x+42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.42 & &\\ & = 169-168 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt1}{2.1} & & = \frac{-13+\sqrt1}{2.1} \\ & = \frac{-14}{2} & & = \frac{-12}{2} \\ & = -7 & & = -6 \\ \\ V &= \Big\{ -7 ; -6 \Big\} & &\end{align} \\ -----------------\)
5. \(16x^2+13x+19=-7x+10\\ \Leftrightarrow 16x^2+20x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+20x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.16.9 & &\\ & = 400-576 & & \\ & = -176 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-12x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.16.49 & &\\ & = 144-3136 & & \\ & = -2992 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(x^2+12x+66=-5x+6\\ \Leftrightarrow x^2+17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt49}{2.1} & & = \frac{-17+\sqrt49}{2.1} \\ & = \frac{-24}{2} & & = \frac{-10}{2} \\ & = -12 & & = -5 \\ \\ V &= \Big\{ -12 ; -5 \Big\} & &\end{align} \\ -----------------\)
8. \(9x^2+27x+12=3x-4\\ \Leftrightarrow 9x^2+24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.9} & & \\ & = -\frac{4}{3} & & \\V &= \Big\{ -\frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.12 & &\\ & = 64-48 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt16}{2.1} & & = \frac{-(-8)+\sqrt16}{2.1} \\ & = \frac{4}{2} & & = \frac{12}{2} \\ & = 2 & & = 6 \\ \\ V &= \Big\{ 2 ; 6 \Big\} & &\end{align} \\ -----------------\)
10. \(16x^2-3x-8=-x-9\\ \Leftrightarrow 16x^2-2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.16.1 & &\\ & = 4-64 & & \\ & = -60 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(x^2-7x+15=-12x+11\\ \Leftrightarrow x^2+5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.1} & & = \frac{-5+\sqrt9}{2.1} \\ & = \frac{-8}{2} & & = \frac{-2}{2} \\ & = -4 & & = -1 \\ \\ V &= \Big\{ -4 ; -1 \Big\} & &\end{align} \\ -----------------\)
12. \(x^2+13x+78=-6x-10\\ \Leftrightarrow x^2+19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.88 & &\\ & = 361-352 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt9}{2.1} & & = \frac{-19+\sqrt9}{2.1} \\ & = \frac{-22}{2} & & = \frac{-16}{2} \\ & = -11 & & = -8 \\ \\ V &= \Big\{ -11 ; -8 \Big\} & &\end{align} \\ -----------------\)