we have
^2 + 12 + 32 = 0
Complete the square
(x^2+12x+6^2-6^2)+32=0
(x^2+12x+36)-36+32=0
(x^2+12x+36)-4=0
REwrite as perfect squares
(x+6)^2=4
take the square root both sides
(x+6)=(+/-)2
x=-6(+/-)2
therefore
x1=-6+2=-4
x2=-6-2=-8
Part 2
we have
2^2 + 11 + 15 = 0
Factor 2
2(x^2+11/2x)+15=0
Complete the square
2(x^2+11/2x+121/16-121/16)+15=0
2(x^2+11/2x+121/16)-121/8+15=0
rewrite as perfect squares
2(x+11/4)^2=121/8-15
2(x+11/4)^2=(121-120)/8
2(x+11/4)^2=1/8
(x+11/4)^2=1/16
take square root both sides
(x+11/4)=(+/-)1/4
x=-11/4(+/-)1/4
therefore
x1=11/4+1/4=12/4=3
x2=11/4-1/4=10/4=2.5
Problem N 3
^2 − 24 + 80 = 0
Complete the square
(x^2-24x+144)-144+80=0
(x-12)^2=144-80
(x-12)^2=64
square root both sides
(x-12)=(+/-)8
x=12(+/-)8
therefore
x1=12+8=20
x2=12-8=4
Problem N 4
^2 − 15 + 36 = 0
(x^2-15x+225/4)-225/4+36=0
(x-15/2)^2=225/4-36
(x-15/2)^2=(225-144)/4
(x-15/2)^2=81/4
square root both sides
(x-15/2)=(+/-)9/2
x=15/2(+/-)9/2
therefore
x1=15/2+19/2=34/2=17
x2=15/2-19/2=-4/2=-2
Problem N 5
4^2 + 16 + 7 = 0
Factor 4
4(x^2+4x)+7=0
4(x^2+4x+4-4)+7=0
4(x^2+4x+4)=16-7
4(x+2)^2=9
(x+2)^2=9/4
square root both sides
(x+2)=(+/-)3/2
x=-2(+/-)3/2
therefore
x1=-2+3/2=1/2=0.5
x2=-2-3/2=-7/2=-3.5