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1 The expression {y}^{2}-2y+2y
2
−2y+2 fits the form a{x}^{2}+bx+cax
2
+bx+c, where:
a=1a=1
b=-2b=−2
c=2c=2
2 Introduce the constant kk, which is 11 in our case
1 Let kk = {(\frac{b}{2a})}^{2}(
2a
b
)
2
From the earlier step, a=1a=1 and b=-2b=−2
Therefore, k={(\frac{b}{2a})}^{2}={(-\frac{2}{2\times 1})}^{2}=1k=(
2a
b
)
2
=(−
2×1
2
)
2
=1
k=1k=1
{y}^{2}-2y+1-1+2y
2
−2y+1−1+2
3 Use Square of Difference: {(a-b)}^{2}={a}^{2}-2ab+{b}^{2}(a−b)
2
=a
2
−2ab+b
2
{(y-1)}^{2}-1+2(y−1)
2
−1+2
4 Simplify
{(y-1)}^{2}+1(y−1)
2
+1
5 Substitute the above back into the original equation
{(y-1)}^{2}+1=0(y−1)
2
+1=0
6 Subtract 11 from both sides
{(y-1)}^{2}=-1(y−1)
2
=−1
7 Take the square root of both sides
y-1=\pm \sqrt{-1}y−1=±√
−1
8 Simplify \sqrt{-1}√
−1
to \sqrt{1}\imath √
1
ı
y-1=\pm \sqrt{1}\imath y−1=±√
1
ı
9 Simplify \sqrt{1}√
1
to 11
y-1=\pm 1\times \imath y−1=±1×ı
10 Simplify 1\times \imath 1×ı to \imath ı
y-1=\pm \imath y−1=±ı
11 Add 11 to both sides
y=\imath +1,-\imath +1y=ı+1,−ı+1
12 Regroup terms
y=1+\imath ,-\imath +1y=1+ı,−ı+1
13 Regroup terms
y=1+\imath ,1-\imath y=1+ı,1−
2
−2y+2 fits the form a{x}^{2}+bx+cax
2
+bx+c, where:
a=1a=1
b=-2b=−2
c=2c=2
2 Introduce the constant kk, which is 11 in our case
1 Let kk = {(\frac{b}{2a})}^{2}(
2a
b
)
2
From the earlier step, a=1a=1 and b=-2b=−2
Therefore, k={(\frac{b}{2a})}^{2}={(-\frac{2}{2\times 1})}^{2}=1k=(
2a
b
)
2
=(−
2×1
2
)
2
=1
k=1k=1
{y}^{2}-2y+1-1+2y
2
−2y+1−1+2
3 Use Square of Difference: {(a-b)}^{2}={a}^{2}-2ab+{b}^{2}(a−b)
2
=a
2
−2ab+b
2
{(y-1)}^{2}-1+2(y−1)
2
−1+2
4 Simplify
{(y-1)}^{2}+1(y−1)
2
+1
5 Substitute the above back into the original equation
{(y-1)}^{2}+1=0(y−1)
2
+1=0
6 Subtract 11 from both sides
{(y-1)}^{2}=-1(y−1)
2
=−1
7 Take the square root of both sides
y-1=\pm \sqrt{-1}y−1=±√
−1
8 Simplify \sqrt{-1}√
−1
to \sqrt{1}\imath √
1
ı
y-1=\pm \sqrt{1}\imath y−1=±√
1
ı
9 Simplify \sqrt{1}√
1
to 11
y-1=\pm 1\times \imath y−1=±1×ı
10 Simplify 1\times \imath 1×ı to \imath ı
y-1=\pm \imath y−1=±ı
11 Add 11 to both sides
y=\imath +1,-\imath +1y=ı+1,−ı+1
12 Regroup terms
y=1+\imath ,-\imath +1y=1+ı,−ı+1
13 Regroup terms
y=1+\imath ,1-\imath y=1+ı,1−
Answered by
3
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