# Solve for z (z+3*y+1)^2+z^2=12*y-4

(z+3⋅y+1)2+z2=12⋅y-4
Rewrite (z+3y+1)2 as (z+3y+1)(z+3y+1).
(z+3y+1)(z+3y+1)+z2=12y-4
Expand (z+3y+1)(z+3y+1) by multiplying each term in the first expression by each term in the second expression.
z⋅z+z(3y)+z⋅1+3yz+3y(3y)+3y⋅1+1z+1(3y)+1⋅1+z2=12y-4
Simplify each term.
Multiply z by z.
z2+z(3y)+z⋅1+3yz+3y(3y)+3y⋅1+1z+1(3y)+1⋅1+z2=12y-4
Rewrite using the commutative property of multiplication.
z2+3zy+z⋅1+3yz+3y(3y)+3y⋅1+1z+1(3y)+1⋅1+z2=12y-4
Multiply z by 1.
z2+3zy+z+3yz+3y(3y)+3y⋅1+1z+1(3y)+1⋅1+z2=12y-4
Multiply y by y.
z2+3zy+z+3yz+3⋅(3y2)+3y⋅1+1z+1(3y)+1⋅1+z2=12y-4
Multiply 3 by 3.
z2+3zy+z+3yz+9y2+3y⋅1+1z+1(3y)+1⋅1+z2=12y-4
Multiply 3 by 1.
z2+3zy+z+3yz+9y2+3y+1z+1(3y)+1⋅1+z2=12y-4
Multiply z by 1.
z2+3zy+z+3yz+9y2+3y+z+1(3y)+1⋅1+z2=12y-4
Multiply 3y by 1.
z2+3zy+z+3yz+9y2+3y+z+3y+1⋅1+z2=12y-4
Multiply 1 by 1.
z2+3zy+z+3yz+9y2+3y+z+3y+1+z2=12y-4
z2+3zy+z+3yz+9y2+3y+z+3y+1+z2=12y-4
Move z.
z2+3yz+3yz+z+9y2+3y+z+3y+1+z2=12y-4
z2+6yz+z+9y2+3y+z+3y+1+z2=12y-4
z2+6yz+z+9y2+3y+z+3y+1+z2=12y-4
z2+6yz+9y2+3y+2z+3y+1+z2=12y-4
z2+6yz+9y2+6y+2z+1+z2=12y-4
2z2+6yz+9y2+6y+2z+1=12y-4
Move all terms to the left side of the equation and simplify.
Move all the expressions to the left side of the equation.
Move 12y to the left side of the equation by subtracting it from both sides.
2z2+6yz+9y2+6y+2z+1-12y=-4
Move 4 to the left side of the equation by adding it to both sides.
2z2+6yz+9y2+6y+2z+1-12y+4=0
2z2+6yz+9y2+6y+2z+1-12y+4=0
Simplify 2z2+6yz+9y2+6y+2z+1-12y+4.
Subtract 12y from 6y.
2z2+6yz+9y2-6y+2z+1+4=0
2z2+6yz+9y2-6y+2z+5=0
2z2+6yz+9y2-6y+2z+5=0
2z2+6yz+9y2-6y+2z+5=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=2, b=6y+2, and c=9y2-6y+5 into the quadratic formula and solve for z.
-(6y+2)±(6y+2)2-4⋅(2⋅(9y2-6y+5))2⋅2
Simplify.
Simplify the numerator.
Apply the distributive property.
z=-(6y)-1⋅2±(6y+2)2-4⋅(2⋅(9y2-6y+5))2⋅2
Multiply 6 by -1.
z=-6y-1⋅2±(6y+2)2-4⋅(2⋅(9y2-6y+5))2⋅2
Multiply -1 by 2.
z=-6y-2±(6y+2)2-4⋅(2⋅(9y2-6y+5))2⋅2
Let u=2⋅(9y2-6y+5). Substitute u for all occurrences of 2⋅(9y2-6y+5).
Rewrite (6y+2)2 as (6y+2)(6y+2).
z=-6y-2±(6y+2)(6y+2)-4⋅u2⋅2
Expand (6y+2)(6y+2) using the FOIL Method.
Apply the distributive property.
z=-6y-2±6y(6y+2)+2(6y+2)-4⋅u2⋅2
Apply the distributive property.
z=-6y-2±6y(6y)+6y⋅2+2(6y+2)-4⋅u2⋅2
Apply the distributive property.
z=-6y-2±6y(6y)+6y⋅2+2(6y)+2⋅2-4⋅u2⋅2
z=-6y-2±6y(6y)+6y⋅2+2(6y)+2⋅2-4⋅u2⋅2
Simplify and combine like terms.
Simplify each term.
Multiply y by y.
z=-6y-2±6⋅(6y2)+6y⋅2+2(6y)+2⋅2-4⋅u2⋅2
Multiply 6 by 6.
z=-6y-2±36y2+6y⋅2+2(6y)+2⋅2-4⋅u2⋅2
Multiply 2 by 6.
z=-6y-2±36y2+12y+2(6y)+2⋅2-4⋅u2⋅2
Multiply 6 by 2.
z=-6y-2±36y2+12y+12y+2⋅2-4⋅u2⋅2
Multiply 2 by 2.
z=-6y-2±36y2+12y+12y+4-4⋅u2⋅2
z=-6y-2±36y2+12y+12y+4-4⋅u2⋅2
z=-6y-2±36y2+24y+4-4⋅u2⋅2
z=-6y-2±36y2+24y+4-4u2⋅2
z=-6y-2±36y2+24y+4-4u2⋅2
Factor 4 out of 36y2+24y+4-4u.
Factor 4 out of 36y2.
z=-6y-2±4(9y2)+24y+4-4u2⋅2
Factor 4 out of 24y.
z=-6y-2±4(9y2)+4(6y)+4-4u2⋅2
Factor 4 out of 4.
z=-6y-2±4(9y2)+4(6y)+4(1)-4u2⋅2
Factor 4 out of -4u.
z=-6y-2±4(9y2)+4(6y)+4(1)+4(-u)2⋅2
Factor 4 out of 4(9y2)+4(6y).
z=-6y-2±4(9y2+6y)+4(1)+4(-u)2⋅2
Factor 4 out of 4(9y2+6y)+4(1).
z=-6y-2±4(9y2+6y+1)+4(-u)2⋅2
Factor 4 out of 4(9y2+6y+1)+4(-u).
z=-6y-2±4(9y2+6y+1-u)2⋅2
z=-6y-2±4(9y2+6y+1-u)2⋅2
Replace all occurrences of u with 2⋅(9y2-6y+5).
z=-6y-2±4(9y2+6y+1-(2⋅(9y2-6y+5)))2⋅2
Simplify.
Simplify each term.
Apply the distributive property.
z=-6y-2±4(9y2+6y+1-(2(9y2)+2(-6y)+2⋅5))2⋅2
Simplify.
Multiply 9 by 2.
z=-6y-2±4(9y2+6y+1-(18y2+2(-6y)+2⋅5))2⋅2
Multiply -6 by 2.
z=-6y-2±4(9y2+6y+1-(18y2-12y+2⋅5))2⋅2
Multiply 2 by 5.
z=-6y-2±4(9y2+6y+1-(18y2-12y+10))2⋅2
z=-6y-2±4(9y2+6y+1-(18y2-12y+10))2⋅2
Apply the distributive property.
z=-6y-2±4(9y2+6y+1-(18y2)-(-12y)-1⋅10)2⋅2
Simplify.
Multiply 18 by -1.
z=-6y-2±4(9y2+6y+1-18y2-(-12y)-1⋅10)2⋅2
Multiply -12 by -1.
z=-6y-2±4(9y2+6y+1-18y2+12y-1⋅10)2⋅2
Multiply -1 by 10.
z=-6y-2±4(9y2+6y+1-18y2+12y-10)2⋅2
z=-6y-2±4(9y2+6y+1-18y2+12y-10)2⋅2
z=-6y-2±4(9y2+6y+1-18y2+12y-10)2⋅2
Subtract 18y2 from 9y2.
z=-6y-2±4(-9y2+6y+1+12y-10)2⋅2
z=-6y-2±4(-9y2+18y+1-10)2⋅2
Subtract 10 from 1.
z=-6y-2±4(-9y2+18y-9)2⋅2
z=-6y-2±4(-9y2+18y-9)2⋅2
Factor 9 out of -9y2+18y-9.
Factor 9 out of -9y2.
z=-6y-2±4(9(-y2)+18y-9)2⋅2
Factor 9 out of 18y.
z=-6y-2±4(9(-y2)+9(2y)-9)2⋅2
Factor 9 out of -9.
z=-6y-2±4(9(-y2)+9(2y)+9(-1))2⋅2
Factor 9 out of 9(-y2)+9(2y).
z=-6y-2±4(9(-y2+2y)+9(-1))2⋅2
Factor 9 out of 9(-y2+2y)+9(-1).
z=-6y-2±4(9(-y2+2y-1))2⋅2
z=-6y-2±4(9(-y2+2y-1))2⋅2
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=-1⋅-1=1 and whose sum is b=2.
Factor 2 out of 2y.
z=-6y-2±4(9(-y2+2(y)-1))2⋅2
Rewrite 2 as 1 plus 1
z=-6y-2±4(9(-y2+(1+1)y-1))2⋅2
Apply the distributive property.
z=-6y-2±4(9(-y2+1y+1y-1))2⋅2
Multiply y by 1.
z=-6y-2±4(9(-y2+y+1y-1))2⋅2
Multiply y by 1.
z=-6y-2±4(9(-y2+y+y-1))2⋅2
z=-6y-2±4(9(-y2+y+y-1))2⋅2
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
z=-6y-2±4(9((-y2+y)+y-1))2⋅2
Factor out the greatest common factor (GCF) from each group.
z=-6y-2±4(9(y(-y+1)-1(-y+1)))2⋅2
z=-6y-2±4(9(y(-y+1)-1(-y+1)))2⋅2
Factor the polynomial by factoring out the greatest common factor, -y+1.
z=-6y-2±4(9((-y+1)(y-1)))2⋅2
z=-6y-2±4(9(-y+1)(y-1))2⋅2
Combine exponents.
Factor -1 out of -y.
z=-6y-2±4(9(-(y)+1)(y-1))2⋅2
Rewrite 1 as -1(-1).
z=-6y-2±4(9(-(y)-1⋅-1)(y-1))2⋅2
Factor -1 out of -(y)-1(-1).
z=-6y-2±4(9(-(y-1))(y-1))2⋅2
Rewrite -(y-1) as -1(y-1).
z=-6y-2±4(9(-1(y-1))(y-1))2⋅2
Raise y-1 to the power of 1.
z=-6y-2±4(9⋅(-1((y-1)(y-1))))2⋅2
Raise y-1 to the power of 1.
z=-6y-2±4(9⋅(-1((y-1)(y-1))))2⋅2
Use the power rule aman=am+n to combine exponents.
z=-6y-2±4(9⋅(-1(y-1)1+1))2⋅2
z=-6y-2±4(9⋅(-1(y-1)2))2⋅2
Multiply 9 by -1.
z=-6y-2±4(-9(y-1)2)2⋅2
z=-6y-2±4⋅(-9(y-1)2)2⋅2
Multiply 4 by -9.
z=-6y-2±-36(y-1)22⋅2
Rewrite -36(y-1)2 as (6(y-1))2⋅-1.
Factor 36 out of -36.
z=-6y-2±36(-1)(y-1)22⋅2
Rewrite 36 as 62.
z=-6y-2±62⋅(-1(y-1)2)2⋅2
Move -1.
z=-6y-2±62(y-1)2⋅-12⋅2
Rewrite 62(y-1)2 as (6(y-1))2.
z=-6y-2±(6(y-1))2⋅-12⋅2
z=-6y-2±(6(y-1))2⋅-12⋅2
Pull terms out from under the radical.
z=-6y-2±6(y-1)-12⋅2
Rewrite -1 as i.
z=-6y-2±6(y-1)i2⋅2
Apply the distributive property.
z=-6y-2±(6y+6⋅-1)i2⋅2
Multiply 6 by -1.
z=-6y-2±(6y-6)i2⋅2
Apply the distributive property.
z=-6y-2±(6yi-6i)2⋅2
z=-6y-2±(6yi-6i)2⋅2
Multiply 2 by 2.
z=-6y-2±(6yi-6i)4
z=-6y-2±(6yi-6i)4
The final answer is the combination of both solutions.
z=3iy-3y-1-3i2
z=-3iy+3y+1-3i2
Solve for z (z+3*y+1)^2+z^2=12*y-4

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