$$ \frac { ( x - 0.03 ) ^ { 2 } } { ( x - 0,04 ) ( x + 0,006 ) } = 60.2 $$

Question

UpStudy AI · Accepted Answer

Para resolver la ecuación dada, primero simplificaremos la expresión y luego resolveremos la ecuación resultante.

Dado que la ecuación es $$ \frac{(x - 0.03)^2}{(x - 0.04)(x + 0.006)} = 60.2 $$, primero simplificaremos la expresión:

$$ \frac{(x - 0.03)^2}{(x - 0.04)(x + 0.006)} = 60.2 $$

Ahora, resolveremos la ecuación resultante.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\frac{\left(x-0.03ight)^{2}}{\left(x-0.04ight)\left(x+0.006ight)}=60.2$$
- step1: Find the domain:
  $$\frac{\left(x-0.03ight)^{2}}{\left(x-0.04ight)\left(x+0.006ight)}=60.2,x \in \left(-\infty,-0.006ight)\cup \left(-0.006,0.04ight)\cup \left(0.04,+\inftyight)$$
- step2: Simplify:
  $$\frac{5\left(100x-3ight)^{2}}{\left(100x-4ight)\left(500x+3ight)}=60.2$$
- step3: Convert the expressions:
  $$\frac{5\left(100x-3ight)^{2}}{\left(100x-4ight)\left(500x+3ight)}=\frac{301}{5}$$
- step4: Cross multiply:
  $$5\left(100x-3ight)^{2}	imes 5=\left(100x-4ight)\left(500x+3ight)	imes 301$$
- step5: Simplify the equation:
  $$25\left(100x-3ight)^{2}=301\left(100x-4ight)\left(500x+3ight)$$
- step6: Calculate:
  $$250000x^{2}-15000x+225=301\left(100x-4ight)\left(500x+3ight)$$
- step7: Calculate:
  $$250000x^{2}-15000x+225=15050000x^{2}-511700x-3612$$
- step8: Move the expression to the left side:
  $$250000x^{2}-15000x+225-\left(15050000x^{2}-511700x-3612ight)=0$$
- step9: Calculate:
  $$-14800000x^{2}+496700x+3837=0$$
- step10: Multiply both sides:
  $$14800000x^{2}-496700x-3837=0$$
- step11: Solve using the quadratic formula:
  $$x=\frac{496700\pm \sqrt{\left(-496700ight)^{2}-4	imes 14800000\left(-3837ight)}}{2	imes 14800000}$$
- step12: Simplify the expression:
  $$x=\frac{496700\pm \sqrt{\left(-496700ight)^{2}-4	imes 14800000\left(-3837ight)}}{29600000}$$
- step13: Simplify the expression:
  $$x=\frac{496700\pm \sqrt{496700^{2}+227150400000}}{29600000}$$
- step14: Simplify the expression:
  $$x=\frac{496700\pm 100\sqrt{4967^{2}+22715040}}{29600000}$$
- step15: Separate into possible cases:
  $$\begin{align}&x=\frac{496700+100\sqrt{4967^{2}+22715040}}{29600000}\&x=\frac{496700-100\sqrt{4967^{2}+22715040}}{29600000}\end{align}$$
- step16: Simplify the expression:
  $$\begin{align}&x=\frac{4967+\sqrt{4967^{2}+22715040}}{296000}\&x=\frac{496700-100\sqrt{4967^{2}+22715040}}{29600000}\end{align}$$
- step17: Simplify the expression:
  $$\begin{align}&x=\frac{4967+\sqrt{4967^{2}+22715040}}{296000}\&x=\frac{4967-\sqrt{4967^{2}+22715040}}{296000}\end{align}$$
- step18: Check if the solution is in the defined range:
  $$\begin{align}&x=\frac{4967+\sqrt{4967^{2}+22715040}}{296000}\&x=\frac{4967-\sqrt{4967^{2}+22715040}}{296000}\end{align},x \in \left(-\infty,-0.006ight)\cup \left(-0.006,0.04ight)\cup \left(0.04,+\inftyight)$$
- step19: Find the intersection:
  $$\begin{align}&x=\frac{4967+\sqrt{4967^{2}+22715040}}{296000}\&x=\frac{4967-\sqrt{4967^{2}+22715040}}{296000}\end{align}$$
- step20: Rewrite:
  $$x_{1}=\frac{4967-\sqrt{4967^{2}+22715040}}{296000},x_{2}=\frac{4967+\sqrt{4967^{2}+22715040}}{296000}$$
La solución de la ecuación $$ \frac{(x - 0.03)^2}{(x - 0.04)(x + 0.006)} = 60.2 $$ es:

$$ x_{1} = \frac{4967 - \sqrt{4967^2 + 22715040}}{296000} $$

$$ x_{2} = \frac{4967 + \sqrt{4967^2 + 22715040}}{296000} $$

Estas son las dos soluciones de la ecuación dada.
Las soluciones de la ecuación son $$ x_{1} $$ y $$ x_{2} $$.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions