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2sec^2(x)-5sec(x)+2=0,0<= x<= 360

Solución

2 sec^{2} (x) - 5 sec (x) + 2 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}

Solución

x = 6 0^{\circ}, x = 30 0^{\circ}

radianes

x = \frac{π}{3}, x = \frac{5 π}{3}

Pasos de solución

2 sec^{2} (x) - 5 sec (x) + 2 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}

Usando el método de sustitución

2 sec^{2} (x) - 5 sec (x) + 2 = 0

Sea:

sec (x) = u

2 u^{2} - 5 u + 2 = 0

2 u^{2} - 5 u + 2 = 0 : u = 2, u = \frac{1}{2}

2 u^{2} - 5 u + 2 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

2 u^{2} - 5 u + 2 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = - 5, c = 2

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

(- 5)^{2} - 4 \cdot 2 \cdot 2 = 3

(- 5)^{2} - 4 \cdot 2 \cdot 2

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 2 \cdot 2

Multiplicar los numeros:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

Restar:

25 - 16 = 9

= 9

Descomponer el número en factores primos:

9 = 3^{2}

= 3^{2}

Aplicar las leyes de los exponentes:

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 5 ) \pm 3}{2 \cdot 2}

Separar las soluciones

u_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 2}

u = \frac{- ( - 5 ) + 3}{2 \cdot 2} : 2

\frac{- ( - 5 ) + 3}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{5 + 3}{2 \cdot 2}

Sumar:

5 + 3 = 8

= \frac{8}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{8}{4}

Dividir:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 5 ) - 3}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 5 ) - 3}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{5 - 3}{2 \cdot 2}

Restar:

5 - 3 = 2

= \frac{2}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2}{4}

Eliminar los terminos comunes:

2

= \frac{1}{2}

Las soluciones a la ecuación de segundo grado son:

u = 2, u = \frac{1}{2}

Sustituir en la ecuación

u = sec (x)

sec (x) = 2, sec (x) = \frac{1}{2}

sec (x) = 2, sec (x) = \frac{1}{2}

sec (x) = 2, 0 \leq x \leq 36 0^{\circ} : x = 6 0^{\circ}, x = 30 0^{\circ}

sec (x) = 2, 0 \leq x \leq 36 0^{\circ}

Soluciones generales para

sec (x) = 2

sec (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Soluciones para el rango

0 \leq x \leq 36 0^{\circ}

x = 6 0^{\circ}, x = 30 0^{\circ}

sec (x) = \frac{1}{2}, 0 \leq x \leq 36 0^{\circ} :

Sin solución

sec (x) = \frac{1}{2}, 0 \leq x \leq 36 0^{\circ}

sec (x) \leq - 1 or sec (x) \geq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 6 0^{\circ}, x = 30 0^{\circ}

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