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

Math
(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.
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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
Add 3zy and 3yz.
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Move z.
z2+3yz+3yz+z+9y2+3y+z+3y+1+z2=12y-4
Add 3yz and 3yz.
z2+6yz+z+9y2+3y+z+3y+1+z2=12y-4
z2+6yz+z+9y2+3y+z+3y+1+z2=12y-4
Add z and z.
z2+6yz+9y2+3y+2z+3y+1+z2=12y-4
Add 3y and 3y.
z2+6yz+9y2+6y+2z+1+z2=12y-4
Add z2 and z2.
2z2+6yz+9y2+6y+2z+1=12y-4
Move all terms to the left side of the equation and simplify.
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Move all the expressions to the left side of the equation.
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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.
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Subtract 12y from 6y.
2z2+6yz+9y2-6y+2z+1+4=0
Add 1 and 4.
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.
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Simplify the numerator.
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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).
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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.
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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.
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Simplify each term.
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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
Add 12y and 12y.
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.
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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.
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Simplify each term.
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Apply the distributive property.
z=-6y-2±4(9y2+6y+1-(2(9y2)+2(-6y)+2⋅5))2⋅2
Simplify.
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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.
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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
Add 6y and 12y.
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.
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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.
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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.
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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.
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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.
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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
Add 1 and 1.
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.
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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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