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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(8-13x)=-48x^2-(5-6x)\)
2. \(x(4x-77)=-121(x+1)\)
3. \(\frac{1}{32}x^2+\frac{1}{4}x+\frac{1}{2}=0\)
4. \((4x-4)(x+2)-x(-5x+30)=-33\)
5. \(\frac{8}{5}x=-\frac{16}{55}x^2-\frac{11}{5}\)
6. \((2x-4)(-x+1)-x(-3x+0)=-1\)
7. \(-(3+24x)=-4x^2-(84-4x)\)
8. \(-(12-52x)=-16x^2-(61-4x)\)
9. \(-\frac{1}{2}x=-\frac{1}{12}x^2-3\)
10. \(-(5-29x)=-x^2-(60-13x)\)
11. \(x(x+11)=3(x-5)\)
12. \(2x^2-(18x-90)=x(x-37)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(8-13x)=-48x^2-(5-6x) \\ \Leftrightarrow -8+13x=-48x^2-5+6x \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
2. \(x(4x-77)=-121(x+1) \\ \Leftrightarrow 4x^2-77x=-121x-121 \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{32}x^2+\frac{1}{4}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{32}x^2+\frac{1}{4}x+\frac{1}{2}\right)=0 \color{red}{.32} \\ \Leftrightarrow x^2+8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.1} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
4. \((4x-4)(x+2)-x(-5x+30)=-33\\ \Leftrightarrow 4x^2+8x-4x-8 +5x^2-30x+33=0 \\ \Leftrightarrow 9x^2-30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-30)}{2.9} & & \\ & = \frac{5}{3} & & \\V &= \Big\{ \frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{8}{5}x=-\frac{16}{55}x^2-\frac{11}{5} \\ \Leftrightarrow \frac{16}{55}x^2+\frac{8}{5}x+\frac{11}{5}=0 \\ \Leftrightarrow \color{red}{55.} \left(\frac{16}{55}x^2+\frac{8}{5}x+\frac{11}{5}\right)=0 \color{red}{.55} \\ \Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-88}{2.16} & & \\ & = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
6. \((2x-4)(-x+1)-x(-3x+0)=-1\\ \Leftrightarrow -2x^2+2x+4x-4 +3x^2+0x+1=0 \\ \Leftrightarrow x^2-2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt16}{2.1} & & = \frac{-(-2)+\sqrt16}{2.1} \\ & = \frac{-2}{2} & & = \frac{6}{2} \\ & = -1 & & = 3 \\ \\ V &= \Big\{ -1 ; 3 \Big\} & &\end{align} \\ -----------------\)
7. \(-(3+24x)=-4x^2-(84-4x) \\ \Leftrightarrow -3-24x=-4x^2-84+4x \\ \Leftrightarrow 4x^2-28x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.81 & &\\ & = 784-1296 & & \\ & = -512 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-(12-52x)=-16x^2-(61-4x) \\ \Leftrightarrow -12+52x=-16x^2-61+4x \\ \Leftrightarrow 16x^2+48x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.49 & &\\ & = 2304-3136 & & \\ & = -832 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(-\frac{1}{2}x=-\frac{1}{12}x^2-3 \\ \Leftrightarrow \frac{1}{12}x^2-\frac{1}{2}x+3=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{1}{2}x+3\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-6x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.36 & &\\ & = 36-144 & & \\ & = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(-(5-29x)=-x^2-(60-13x) \\ \Leftrightarrow -5+29x=-x^2-60+13x \\ \Leftrightarrow x^2+16x+55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.55 & &\\ & = 256-220 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt36}{2.1} & & = \frac{-16+\sqrt36}{2.1} \\ & = \frac{-22}{2} & & = \frac{-10}{2} \\ & = -11 & & = -5 \\ \\ V &= \Big\{ -11 ; -5 \Big\} & &\end{align} \\ -----------------\)
11. \(x(x+11)=3(x-5) \\ \Leftrightarrow x^2+11x=3x-15 \\ \Leftrightarrow x^2+8x+15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.15 & &\\ & = 64-60 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt4}{2.1} & & = \frac{-8+\sqrt4}{2.1} \\ & = \frac{-10}{2} & & = \frac{-6}{2} \\ & = -5 & & = -3 \\ \\ V &= \Big\{ -5 ; -3 \Big\} & &\end{align} \\ -----------------\)
12. \(2x^2-(18x-90)=x(x-37) \\ \Leftrightarrow 2x^2-18x+90=x^2-37x \\ \Leftrightarrow x^2+19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.90 & &\\ & = 361-360 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt1}{2.1} & & = \frac{-19+\sqrt1}{2.1} \\ & = \frac{-20}{2} & & = \frac{-18}{2} \\ & = -10 & & = -9 \\ \\ V &= \Big\{ -10 ; -9 \Big\} & &\end{align} \\ -----------------\)