Question

determine as raizes  reais das equações
abaixo usando o metodo de complemento de quadrado.

a) x²+8x+15=0
b) y²-2y-3=0
c) x²-14x+50=0

Answer 1

a)

$x^{2}+8x+15=0\\x^{2}+8x+15+1-1=0\\x^{2}+8x+16-1=0\\(x+4)^{2}-1=0\\(x+4)^{2}-1^{2}=0$

Sabe-se que (a + b)(a - b) = a² - b²:

$(x+4+1)(x+4-1)=0\\(x+5)(x+3)=0$

Igualando as 2 parcelas a zero:

$x+3=0\\x=-3\\\\x+5=0\\x=-5$

S = {-5,-3}
___________________________

b)

$y^{2}-2y-3=0\\y^{2}-2y-3+4-4=0\\y^{2}-2y+1-4=0\\(y-1)^{2}-4=0\\(y-1)^{2}-2^{2}=0\\(y-1+2)(y-1-2)=0\\(y+1)(y-3)=0$

$y+1=0\\y=-1\\\\y-3=0\\y=3$

S = {-1,3}
___________________________

c)

$x^{2}-14x+50=0\\x^{2}-14x+50-1+1=0\\x^{2}-14x+49+1=0\\(x-7)^{2}+1=0\\(x-7)^{2}-(-1)=0\\(x-7)^{2}-(\sqrt{-1})^{2}=0\\(x-7)^{2}-i^{2}=0\\(x-7+i)(x-7-i)=0$

$x-7+i=0\\x=7-i\\\\x-7-i=0\\x=7+i$

S = {7 - i,7 + i}