Question

VALE 40 PONTOS

resolva, em R, as seguintes equações:

$4 { x}^{2} - 12x + 9 = 0$

$9x {}^{2} - 24x + 16 = 0$

$x { }^{2} - 12x + 3 = 0$

$x {}^{2} - x - 2 = 0$

$- {x}^{2} + 2x + 15 = 0$

$4x {}^{4} - 5x {}^{2} + 1 = 0$

${x}^{4} - 8 {x}^{2} + 15 = 0$

${x}^{4} - 36 {x}^{2} = 0$

$\sqrt{5x + 9} = x - 1$

$\sqrt{8x + 4} = \sqrt{5x + 16}$
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Answer 1

Olá!

a) 4x² - 12x + 9 =0

(2x -3)* (2x-3) = 0

2x- 3= 0

2X= 3

X= 3/2

Solução (3/2)

b) 9X² -24X+16 =0

(3X - 4)*(3X-4) =0

3X- 4= 0

3X= 4

X= 4/3

Solução: (4/3)

c) X² -12X + 3 =0

a=1

b= -12

c= 3

Δ= b² -4ac

Δ= (-12)² - 4 * 1 *3

Δ= 144 - 12

Δ= 132

x= -b±√Δ/2a

x= (-(-12) ±√132)/ 2*1

x= (12±2√33)/ 2

x'= (12+2√33)/2

x'= 6+√33

x"= (12-2√33)/2

x"= 6 -√33

Solução: (6+√33; 6-√33)

d)x² -x -2= 0

(x-2)*(x+1)=0

x-2= 0

x= 2

x+1= 0

x= -1

Solução: (-1,2)

e) -x²+2x+15=0

(-x-3)*(x-5)=0

-x-3= 0

x= -3

x-5= 0

x=5

Solução: (-3,5)

f) 4(x²)² -5x²+1=0

x²= y

4y² - 5y+1 = 0

(4y-1)*(y-1)=0

4y- 1= 0

4y= 1

y= 1/4

y-1= 0

y=1

x²= y

x²= 1

x= ±√1

x= ±1

x²=y

x² = (1/4)

x= ±√(1/4)

x= ±1/2

Solução: (1/2 ; -1/2; -1; 1)

g) (x²)²-8x²+15

x² =y

y² - 8y + 15

(y-3)*(y-5)=0

y-3= 0

y= 3

y-5= 0

y=5

x²= y

x²= 3

x= ±√3

x²= y

x²= 5

x= ±√5

Solução: (-√5, -√3, √5, √3)

h) (x²)² -36x² =0

x²=y

y² - 36y = 0

y*(y- 36)=0

y= 0

y-36=0

y= 36

x²=y

x²= 0

x=√0

x= 0

x²= y

x²=36

x= ±√36

x= ±6

Solução: (-6, 0 , 6)

i) (√(5x+9))² = (x-1)²

5x +9= x² - 2x +1

-x²+7x+8= 0

a= -1

b= 7

c=8

Δ= b²-4ac

Δ= 7² - 4*-1*8

Δ= 49 +32

Δ= 81

x= -b±√Δ/2a

x= (-7 ±√81)/ 2*-1

x= (-7±9/ -2

x'= (-7+9)/-2

X'= 2/-2= -1

x"= (-7-9)/-2

X"= -16/-2

X"=8

Solução: (-1,8)

j) √(8x+4)² = √(5x+16)²

8x+4= x+16

8x-x= 16-4

7x= 12

x= 12/7

solução: (12/7)

Bons estudos!

T.H.S.G.

Answer 2

4x²—12x+9=0

∆=(-12)²—4•4•9

∆=144—144

∆=0

x'ex'' = (12±0)/2•4

x'ex''=12/8=3/2

B)

9x²—24x+16=0

∆=(-24)²—4•9•16

∆=576—576

∆=0.

x'ex''=(24±0)/2•9

x'ex''=24/18

x'ex'' = 4/3

C)

x²—12x+3=0

∆=(-12)²—4•1•3

∆=144—12

∆=136

x'ex'' = (12±√136)/2•1

x'ex'' = (12±√136)/2

x'ex'' = 6±√136

D)

x²—x—2=0

∆=(-1)²—4•1•(-2)

∆=1+8

∆=9

x'ex'' = ( 1±√9)/2•1

x'ex'' = (1±3)/2

x' = (1+3)/2 = 4/2 = 2

x'' = (1-3)/2 = -2/2 = -1

E)

—x²+2x+15=0

∆=2²—4•(-1)•15

∆=4+60

∆=64

x' = (-2+8)/2•(-1)

x' = 6/-2 = -3

x'' = (-2-8)/2•(-1)

x'' = -10/-2 = 5

F)

4x⁴—5x²+1=0

4(x²)²—5x²+1=0

Seja: x²=t

4t²—5t+1=0

∆ = (-5)²—4•4•1

∆ = 25—16

∆ = 9

t' = ( 5+3)/2•4

t'= 8/8 = 1

t'' = (5-3)/2•4

t'' = 2/8 = 1/4

x'ex''=±√t'. /\ x'''ex''''=±√t''

x'ex''=±√1=±1/\x'''ex''''=±√1/4

G)

x⁴—8x²+15=0

(x²)²—8x²+15=0

Seja x²=t

t²—8t+15=0

∆=(-8)²—4•1•15

∆=64—60

∆=4

t' = ( 8+2)/2•1

t' = 10/2 = 5

t'' = (8-2)/2•1

t'' = 6/2 = 3

x' e x'' = ±√t' /\ x''' e x''''=±√t''

x' e x'' = ±√5/\ x''' e x''''=±√3

H)

x⁴—36x²=0

(x²)²—3x²=0

Seja: x²=t

t²—36t=0

t(t—36)=0

t=0 /\ t=36

x' e x''=±√t'. /\x'''ex''''=±√t''

x'ex''=±√0=0 /\x'''ex''''=±√36=±6

I)

√(5x+9)=x—1

[√(5x+9)]² = (x—5)²

5x+9 = x²—2•x•5+5²

5x+9 = x²—10x+25

x²—10x+25=5x+9

x²—10x—5x+25—9=0

x²—15x+16=0

∆=(-15)²—4•1•16

∆=225—64

∆=161

x' e x''= ( 15±√161)/2•1

x'ex'' = (15±√161)/2

I)

√(8x+4)=√(5x+16)

[√(8x+4)]²=[√(5x+16)]²

8x+4 = 5x+16

8x—5x = 16—4

3x = 12

x = 12/3

x = 4