Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(9x^2+30x+25=0\)
- \(6x^2+37x+13=12x-11\)
- \(9x^2+47x+76=3x+12\)
- \(x^2-24x+144=0\)
- \(4x^2+9x+3=-8x-1\)
- \(9x^2+28x+25=0\)
- \(9x^2-23x+28=x+12\)
- \(12x^2+31x+21=6x+9\)
- \(x^2+4x+4=0\)
- \(x^2+18x+80=0\)
- \(x^2+12x+36=0\)
- \(9x^2-7x-13=-12x-9\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+30x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (30)^2-4.9.25 & &\\
& = 900-900 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-30}{2.9} & & \\
& = -\frac{5}{3} & & \\V &= \Big\{ -\frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
- \(6x^2+37x+13=12x-11\\
\Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.6.24 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\
& = \frac{-32}{12} & & = \frac{-18}{12} \\
& = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
- \(9x^2+47x+76=3x+12\\
\Leftrightarrow 9x^2+44x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+44x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (44)^2-4.9.64 & &\\
& = 1936-2304 & & \\
& = -368 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.1.144 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.1} & & \\
& = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)
- \(4x^2+9x+3=-8x-1\\
\Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.4.4 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\
& = \frac{-32}{8} & & = \frac{-2}{8} \\
& = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+28x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (28)^2-4.9.25 & &\\
& = 784-900 & & \\
& = -116 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(9x^2-23x+28=x+12\\
\Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.9.16 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.9} & & \\
& = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
- \(12x^2+31x+21=6x+9\\
\Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.12.12 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\
& = \frac{-32}{24} & & = \frac{-18}{24} \\
& = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (4)^2-4.1.4 & &\\
& = 16-16 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-4}{2.1} & & \\
& = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+80=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (18)^2-4.1.80 & &\\
& = 324-320 & & \\
& = 4 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-18-\sqrt4}{2.1} & & = \frac{-18+\sqrt4}{2.1} \\
& = \frac{-20}{2} & & = \frac{-16}{2} \\
& = -10 & & = -8 \\ \\ V &= \Big\{ -10 ; -8 \Big\} & &\end{align} \\ -----------------\)
- \(\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (12)^2-4.1.36 & &\\
& = 144-144 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-12}{2.1} & & \\
& = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
- \(9x^2-7x-13=-12x-9\\
\Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.9.(-4) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\
& = \frac{-18}{18} & & = \frac{8}{18} \\
& = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)