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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(9-15x)=-x^2-(21-2x)\)
2. \(-(13-42x)=-9x^2-(38-20x)\)
3. \(\frac{1}{2}x^2+\frac{5}{2}x-12=0\)
4. \((2x-5)(-3x-4)-x(-7x+0)=-12\)
5. \(12x^2-(20x-30)=11x(x-3)\)
6. \(x(6x+16)=3(x-2)\)
7. \(2x^2-(3x+12)=x(x-7)\)
8. \(-(6-17x)=-2x^2-(-12-12x)\)
9. \(2x^2-(9x-64)=x(x+7)\)
10. \(-(15-23x)=-x^2-(36-13x)\)
11. \(x(x+1)=9(x+1)\)
12. \(x(4x+25)=5(x-5)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(9-15x)=-x^2-(21-2x) \\ \Leftrightarrow -9+15x=-x^2-21+2x \\ \Leftrightarrow x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.12 & &\\ & = 169-48 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt121}{2.1} & & = \frac{-13+\sqrt121}{2.1} \\ & = \frac{-24}{2} & & = \frac{-2}{2} \\ & = -12 & & = -1 \\ \\ V &= \Big\{ -12 ; -1 \Big\} & &\end{align} \\ -----------------\)
2. \(-(13-42x)=-9x^2-(38-20x) \\ \Leftrightarrow -13+42x=-9x^2-38+20x \\ \Leftrightarrow 9x^2+22x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+22x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.9.25 & &\\ & = 484-900 & & \\ & = -416 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(\frac{1}{2}x^2+\frac{5}{2}x-12=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{5}{2}x-12\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-24) & &\\ & = 25+96 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt121}{2.1} & & = \frac{-5+\sqrt121}{2.1} \\ & = \frac{-16}{2} & & = \frac{6}{2} \\ & = -8 & & = 3 \\ \\ V &= \Big\{ -8 ; 3 \Big\} & &\end{align} \\ -----------------\)
4. \((2x-5)(-3x-4)-x(-7x+0)=-12\\ \Leftrightarrow -6x^2-8x+15x+20 +7x^2+0x+12=0 \\ \Leftrightarrow x^2+12x+32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt16}{2.1} & & = \frac{-12+\sqrt16}{2.1} \\ & = \frac{-16}{2} & & = \frac{-8}{2} \\ & = -8 & & = -4 \\ \\ V &= \Big\{ -8 ; -4 \Big\} & &\end{align} \\ -----------------\)
5. \(12x^2-(20x-30)=11x(x-3) \\ \Leftrightarrow 12x^2-20x+30=11x^2-33x \\ \Leftrightarrow x^2+13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt49}{2.1} & & = \frac{-13+\sqrt49}{2.1} \\ & = \frac{-20}{2} & & = \frac{-6}{2} \\ & = -10 & & = -3 \\ \\ V &= \Big\{ -10 ; -3 \Big\} & &\end{align} \\ -----------------\)
6. \(x(6x+16)=3(x-2) \\ \Leftrightarrow 6x^2+16x=3x-6 \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
7. \(2x^2-(3x+12)=x(x-7) \\ \Leftrightarrow 2x^2-3x-12=x^2-7x \\ \Leftrightarrow x^2+4x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-12) & &\\ & = 16+48 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt64}{2.1} & & = \frac{-4+\sqrt64}{2.1} \\ & = \frac{-12}{2} & & = \frac{4}{2} \\ & = -6 & & = 2 \\ \\ V &= \Big\{ -6 ; 2 \Big\} & &\end{align} \\ -----------------\)
8. \(-(6-17x)=-2x^2-(-12-12x) \\ \Leftrightarrow -6+17x=-2x^2+12+12x \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
9. \(2x^2-(9x-64)=x(x+7) \\ \Leftrightarrow 2x^2-9x+64=x^2+7x \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
10. \(-(15-23x)=-x^2-(36-13x) \\ \Leftrightarrow -15+23x=-x^2-36+13x \\ \Leftrightarrow x^2+10x+21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.21 & &\\ & = 100-84 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt16}{2.1} & & = \frac{-10+\sqrt16}{2.1} \\ & = \frac{-14}{2} & & = \frac{-6}{2} \\ & = -7 & & = -3 \\ \\ V &= \Big\{ -7 ; -3 \Big\} & &\end{align} \\ -----------------\)
11. \(x(x+1)=9(x+1) \\ \Leftrightarrow x^2+x=9x+9 \\ \Leftrightarrow x^2-8x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-9) & &\\ & = 64+36 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt100}{2.1} & & = \frac{-(-8)+\sqrt100}{2.1} \\ & = \frac{-2}{2} & & = \frac{18}{2} \\ & = -1 & & = 9 \\ \\ V &= \Big\{ -1 ; 9 \Big\} & &\end{align} \\ -----------------\)
12. \(x(4x+25)=5(x-5) \\ \Leftrightarrow 4x^2+25x=5x-25 \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)