Question

Equações completas do 2° grau

A) (x - 1) (x - 2) = 6

B) 2x (x + 5) = x² + 5x + 14

C) (x + 6) (x - 3) = 6x - 17

D) (x + 1) (2x + 3) = 4x² - 22

Answer 1

A)
(x-1)(x-2)=6
x²-2x-1x+2=6
x²-3x+2-6=0
x²-3x-4=0
Δ=(-3)² - 4·1·(-4)
Δ=9 + 16
Δ=25
$x=\frac{-(-3)\pm \sqrt{25}}{2\cdot 1}\to x=\frac{3\pm 5}{2}\\ \\ x'=\frac{8}{2}\to \boxed{x'=4}\\ \\ x''=\frac{-2}{2}\to \boxed{x''=-1}$
B)
2x(x+5)=x²+5x+14
2x²+10x=x²+5x+14
x²+5x+14-2x²-10x=0
-x²-5x+14=0
Δ=(-5)² - 4·(-1)·14
Δ=25 + 56
Δ=81
$x=\frac{-(-5)\pm \sqrt{81}}{2\cdot (-1)}\to x=\frac{5\pm 9}{-2}\\ \\ x'=\frac{14}{-2}\to \boxed{x'=-7}\\ \\ x''=\frac{-4}{-2}\to \boxed{x''=2}$
C)
(x+6)(x-3)=6x-17
x²-3x+6x-18=6x-17
x²-3x+6x-18-6x+17=0
x²-3x-1=0
Δ=(-3)² - 4·1·(-1)
Δ=9 + 4
Δ=13
$x=\frac{-(-3)\pm \sqrt{13}}{2\cdot 1}\to x=\frac{3\pm \sqrt{13}}{2}\\ \\ \boxed{x'=\frac{(3+\sqrt{13})}{2}}\\ \\ \boxed{x''=\frac{3-\sqrt{13}}{2}}$
D)
(x+1)(2x+3)=4x²-22
2x²+3x+2x+3=4x²-22
2x²+3x+2x+3-4x²+22=0
-2x²+5x+25=0
Δ=5² - 4·(-2)·25
Δ=25 + 200
Δ=225
$x=\frac{-5\pm \sqrt{225}}{2\cdot (-2)}\to x=\frac{-5\pm 15}{-4}\\ \\ x'=\frac{10}{-4}\to \boxed{x'= -2,5}\\ \\ x''=\frac{-20}{-4}\to \boxed{x''=5}$

Espero ter ajudado.

Answer 2

Resposta:

Explicação passo-a-passo:

A)

(x-1)(x-2)=6

x²-2x-1x+2=6

x²-3x+2-6=0

x²-3x-4=0

Δ=(-3)² - 4·1·(-4)

Δ=9 + 16

Δ=25

B)

2x(x+5)=x²+5x+14

2x²+10x=x²+5x+14

x²+5x+14-2x²-10x=0

-x²-5x+14=0

Δ=(-5)² - 4·(-1)·14

Δ=25 + 56

Δ=81

C)

(x+6)(x-3)=6x-17

x²-3x+6x-18=6x-17

x²-3x+6x-18-6x+17=0

x²-3x-1=0

Δ=(-3)² - 4·1·(-1)

Δ=9 + 4

Δ=13

D)

(x+1)(2x+3)=4x²-22

2x²+3x+2x+3=4x²-22

2x²+3x+2x+3-4x²+22=0

-2x²+5x+25=0

Δ=5² - 4·(-2)·25

Δ=25 + 200

Δ=225

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