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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{4}x^2+\frac{23}{4}x+33=0\)
2. \((-5x+1)(5x-4)-x(-33x+9)=14\)
3. \(-(10-7x)=-x^2-(8-6x)\)
4. \(\frac{1}{3}x^2+\frac{13}{3}x+\frac{40}{3}=0\)
5. \(-(9-17x)=-16x^2-(73-9x)\)
6. \(x(16x-14)=2(x-2)\)
7. \((x-3)(-2x+3)-x(-5x-31)=-57\)
8. \(17x^2-(15x-36)=x(x+33)\)
9. \(\frac{25}{12}x=-\frac{1}{3}x^2-3\)
10. \(\frac{1}{15}x^2-\frac{2}{3}x+\frac{5}{3}=0\)
11. \((x+4)(-2x+3)-x(-6x-21)=-69\)
12. \(2x^2-(17x-96)=x(x-37)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{4}x^2+\frac{23}{4}x+33=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{23}{4}x+33\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+23x+132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+23x+132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (23)^2-4.1.132 & &\\ & = 529-528 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-23-\sqrt1}{2.1} & & = \frac{-23+\sqrt1}{2.1} \\ & = \frac{-24}{2} & & = \frac{-22}{2} \\ & = -12 & & = -11 \\ \\ V &= \Big\{ -12 ; -11 \Big\} & &\end{align} \\ -----------------\)
2. \((-5x+1)(5x-4)-x(-33x+9)=14\\ \Leftrightarrow -25x^2+20x+5x-4 +33x^2-9x-14=0 \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
3. \(-(10-7x)=-x^2-(8-6x) \\ \Leftrightarrow -10+7x=-x^2-8+6x \\ \Leftrightarrow x^2+x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt9}{2.1} & & = \frac{-1+\sqrt9}{2.1} \\ & = \frac{-4}{2} & & = \frac{2}{2} \\ & = -2 & & = 1 \\ \\ V &= \Big\{ -2 ; 1 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{3}x^2+\frac{13}{3}x+\frac{40}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{13}{3}x+\frac{40}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+13x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.40 & &\\ & = 169-160 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt9}{2.1} & & = \frac{-13+\sqrt9}{2.1} \\ & = \frac{-16}{2} & & = \frac{-10}{2} \\ & = -8 & & = -5 \\ \\ V &= \Big\{ -8 ; -5 \Big\} & &\end{align} \\ -----------------\)
5. \(-(9-17x)=-16x^2-(73-9x) \\ \Leftrightarrow -9+17x=-16x^2-73+9x \\ \Leftrightarrow 16x^2+8x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.16.64 & &\\ & = 64-4096 & & \\ & = -4032 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(x(16x-14)=2(x-2) \\ \Leftrightarrow 16x^2-14x=2x-4 \\ \Leftrightarrow 16x^2-16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.16} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
7. \((x-3)(-2x+3)-x(-5x-31)=-57\\ \Leftrightarrow -2x^2+3x+6x-9 +5x^2+31x+57=0 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
8. \(17x^2-(15x-36)=x(x+33) \\ \Leftrightarrow 17x^2-15x+36=x^2+33x \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{25}{12}x=-\frac{1}{3}x^2-3 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{25}{12}x+3=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2+\frac{25}{12}x+3\right)=0 \color{red}{.12} \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{1}{15}x^2-\frac{2}{3}x+\frac{5}{3}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2-\frac{2}{3}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow x^2-10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-10)}{2.1} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
11. \((x+4)(-2x+3)-x(-6x-21)=-69\\ \Leftrightarrow -2x^2+3x-8x+12 +6x^2+21x+69=0 \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(2x^2-(17x-96)=x(x-37) \\ \Leftrightarrow 2x^2-17x+96=x^2-37x \\ \Leftrightarrow x^2+20x+96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.96 & &\\ & = 400-384 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-20-\sqrt16}{2.1} & & = \frac{-20+\sqrt16}{2.1} \\ & = \frac{-24}{2} & & = \frac{-16}{2} \\ & = -12 & & = -8 \\ \\ V &= \Big\{ -12 ; -8 \Big\} & &\end{align} \\ -----------------\)