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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((2x+3)(4x+1)-x(7x+10)=-3\)
2. \(\frac{1}{2}x^2+\frac{13}{6}x+2=0\)
3. \(\frac{13}{9}x=-\frac{4}{3}x^2-\frac{1}{3}\)
4. \(\frac{25}{8}x=-\frac{9}{2}x^2-\frac{1}{2}\)
5. \(\frac{1}{7}x^2+x+\frac{7}{4}=0\)
6. \(-(11-26x)=-16x^2-(36-16x)\)
7. \(2x^2-(10x+84)=x(x-15)\)
8. \(11x^2-(6x-121)=2x(x-36)\)
9. \((-5x+5)(-4x-2)-x(11x-72)=-154\)
10. \(-(2+5x)=-x^2-(-6-2x)\)
11. \(-(8-29x)=-8x^2-(10-12x)\)
12. \(\frac{1}{2}x=-\frac{1}{4}x^2+\frac{15}{4}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((2x+3)(4x+1)-x(7x+10)=-3\\ \Leftrightarrow 8x^2+2x+12x+3 -7x^2-10x+3=0 \\ \Leftrightarrow x^2-5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt1}{2.1} & & = \frac{-(-5)+\sqrt1}{2.1} \\ & = \frac{4}{2} & & = \frac{6}{2} \\ & = 2 & & = 3 \\ \\ V &= \Big\{ 2 ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{2}x^2+\frac{13}{6}x+2=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{2}x^2+\frac{13}{6}x+2\right)=0 \color{red}{.6} \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{13}{9}x=-\frac{4}{3}x^2-\frac{1}{3} \\ \Leftrightarrow \frac{4}{3}x^2+\frac{13}{9}x+\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{9.} \left(\frac{4}{3}x^2+\frac{13}{9}x+\frac{1}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{25}{8}x=-\frac{9}{2}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{9}{2}x^2+\frac{25}{8}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{2}x^2+\frac{25}{8}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{7}x^2+x+\frac{7}{4}=0\\ \Leftrightarrow \color{red}{28.} \left(\frac{1}{7}x^2+x+\frac{7}{4}\right)=0 \color{red}{.28} \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(-(11-26x)=-16x^2-(36-16x) \\ \Leftrightarrow -11+26x=-16x^2-36+16x \\ \Leftrightarrow 16x^2+10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.25 & &\\ & = 100-1600 & & \\ & = -1500 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(2x^2-(10x+84)=x(x-15) \\ \Leftrightarrow 2x^2-10x-84=x^2-15x \\ \Leftrightarrow x^2+5x-84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-84) & &\\ & = 25+336 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt361}{2.1} & & = \frac{-5+\sqrt361}{2.1} \\ & = \frac{-24}{2} & & = \frac{14}{2} \\ & = -12 & & = 7 \\ \\ V &= \Big\{ -12 ; 7 \Big\} & &\end{align} \\ -----------------\)
8. \(11x^2-(6x-121)=2x(x-36) \\ \Leftrightarrow 11x^2-6x+121=2x^2-72x \\ \Leftrightarrow 9x^2+66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-66}{2.9} & & \\ & = -\frac{11}{3} & & \\V &= \Big\{ -\frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
9. \((-5x+5)(-4x-2)-x(11x-72)=-154\\ \Leftrightarrow 20x^2+10x-20x-10 -11x^2+72x+154=0 \\ \Leftrightarrow 9x^2+72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.9} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
10. \(-(2+5x)=-x^2-(-6-2x) \\ \Leftrightarrow -2-5x=-x^2+6+2x \\ \Leftrightarrow x^2-7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-8) & &\\ & = 49+32 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt81}{2.1} & & = \frac{-(-7)+\sqrt81}{2.1} \\ & = \frac{-2}{2} & & = \frac{16}{2} \\ & = -1 & & = 8 \\ \\ V &= \Big\{ -1 ; 8 \Big\} & &\end{align} \\ -----------------\)
11. \(-(8-29x)=-8x^2-(10-12x) \\ \Leftrightarrow -8+29x=-8x^2-10+12x \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{1}{2}x=-\frac{1}{4}x^2+\frac{15}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{1}{2}x-\frac{15}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{1}{2}x-\frac{15}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-15) & &\\ & = 4+60 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt64}{2.1} & & = \frac{-2+\sqrt64}{2.1} \\ & = \frac{-10}{2} & & = \frac{6}{2} \\ & = -5 & & = 3 \\ \\ V &= \Big\{ -5 ; 3 \Big\} & &\end{align} \\ -----------------\)