Question

resuelve las siguientes ecuaciones y expresa el resultado en forma de numeros complejos
1.- 3X2+8X+12=0
2.- 4X2+20X+31=0
3.-2X2-12x+21=0
4.-7x2+12x+8=0
5.-y2+14y+60=0
6.- x2-2ax+a2+b2=0

Answer 1

Respuesta:

Explicación paso a paso:

Apliquemos la formula general a cada ecuación

$1.-\mathbf{3x^{2}+8x+12=0}\\\\\mathbf{a=3, b=8, c=12}\\\\\mathsf{x=\frac{-8\pm\sqrt{8^{2}-4(3)(12)}}{2(3)}}\\\\\mathsf{x=\frac{-8\pm\sqrt{64-144}}{6}}\\\\\mathsf{x=\frac{-8\pm\sqrt{-80}}{6}}\\\\\mathsf{x=\frac{-8\pm2\sqrt{20}i}{6}}\\\\\mathbf{x_{1}=-\frac{4}{3}+\frac{\sqrt{20}i}{3}}\\\\\mathbf{x_{2}=-\frac{4}{3}-\frac{\sqrt{20}i}{3}}\\\\2.-\mathbf{4x^{2}+20x+31=0}\\\\\mathbf{a=4, b=20, c,=31}\\\\\mathsf{x=\frac{-20\pm\sqrt{(20)^{2}-4(4)(31)}}{2(4)}}\\\\\mathsf{\frac{-20\pm\sqrt{400-496}}{8}}\\\\\mathsf{\frac{-20\pm\sqrt{-96}}{8}}\\\\\mathsf{\frac{-20\pm4\sqrt{6}i}{8}}=-\frac{5\pm\sqrt{6}i}{2}}\\\\\mathbf{x_{1}=-\frac{5}{2}+\frac{\sqrt{6}i}{2}}\\\\\mathbf{x_{2}=-\frac{5}{2}-\frac{\sqrt{6}i}{2}}$

3.-

$\mathbf{2x^{2}-12x+21=0}\\\\\mathbf{a=2, b=-12, c=21}\\\\\mathsf{x=\frac{12\pm\sqrt{(-12)^{2}-4(2)(21)}}{2(2)}}\\\\\mathsf{x=\frac{12\pm\sqrt{144-168}}{4}}\\\\\mathsf{x=\frac{12\pm\sqrt{-24}}{4}}\\\\\mathsf{x=\frac{12\pm2\sqrt{6}i}{4}}\\\\\mathbf{x_{1}=3+\frac{\sqrt{6}i}{2}}\\\\\mathbf{x_{2}=3-\frac{\sqrt{6}i}{2}}$

4.-

$\mathbf{x_{1}=-\frac{6}{7}+\frac{2\sqrt{5}i}{7}}\\\\\mathbf{x_{2}=-\frac{6}{7}-\frac{2\sqrt{5}i}{7}}$

6.-

$x_{1}=-7+2\sqrt{6}i\\\\x_{2}=-7-2\sqrt{6}i$

7.-

$x^{2}+2a+a^{2}+b^{2}=0\\\\a=1, b=2a, c=a^{2}+b^{2}\\\\x=\frac{-2a\pm\sqrt{(2a)^{2}-4(1)(a^{2}+b^{2})}}{2(1)}\\\\x=\frac{-2a\pm\sqrt{4a^{2}-4a^{2}-4b^{2}}}{2}\\\\x=\frac{-2a\pm\sqrt{-4b^{2}}}{2}\\\\x=\frac{-2a\pm 2bi}{2}\\\\x=-a\pm bi\\\\x_{1}=-a+bi\\\\x_{2}=-a-bi$

Saludos.