# Solve for y square root of 4y+5- square root of y-1=3

4y+5-y-1=3
Add y-1 to both sides of the equation.
4y+5=3+y-1
To remove the radical on the left side of the equation, square both sides of the equation.
4y+52=(3+y-1)2
Simplify each side of the equation.
Multiply the exponents in ((4y+5)12)2.
Apply the power rule and multiply exponents, (am)n=amn.
(4y+5)12⋅2=(3+y-1)2
Cancel the common factor of 2.
Cancel the common factor.
(4y+5)12⋅2=(3+y-1)2
Rewrite the expression.
(4y+5)1=(3+y-1)2
(4y+5)1=(3+y-1)2
(4y+5)1=(3+y-1)2
Simplify.
4y+5=(3+y-1)2
4y+5=(3+y-1)2
Solve for y.
Simplify (3+y-1)2.
Rewrite (3+y-1)2 as (3+y-1)(3+y-1).
4y+5=(3+y-1)(3+y-1)
Expand (3+y-1)(3+y-1) using the FOIL Method.
Apply the distributive property.
4y+5=3(3+y-1)+y-1(3+y-1)
Apply the distributive property.
4y+5=3⋅3+3y-1+y-1(3+y-1)
Apply the distributive property.
4y+5=3⋅3+3y-1+y-1⋅3+y-1y-1
4y+5=3⋅3+3y-1+y-1⋅3+y-1y-1
Simplify and combine like terms.
Simplify each term.
Multiply 3 by 3.
4y+5=9+3y-1+y-1⋅3+y-1y-1
Move 3 to the left of y-1.
4y+5=9+3y-1+3⋅y-1+y-1y-1
Multiply y-1y-1.
Raise y-1 to the power of 1.
4y+5=9+3y-1+3y-1+y-11y-1
Raise y-1 to the power of 1.
4y+5=9+3y-1+3y-1+y-11y-11
Use the power rule aman=am+n to combine exponents.
4y+5=9+3y-1+3y-1+y-11+1
4y+5=9+3y-1+3y-1+y-12
4y+5=9+3y-1+3y-1+y-12
Rewrite y-12 as y-1.
Use axn=axn to rewrite y-1 as (y-1)12.
4y+5=9+3y-1+3y-1+((y-1)12)2
Apply the power rule and multiply exponents, (am)n=amn.
4y+5=9+3y-1+3y-1+(y-1)12⋅2
Combine 12 and 2.
4y+5=9+3y-1+3y-1+(y-1)22
Cancel the common factor of 2.
Cancel the common factor.
4y+5=9+3y-1+3y-1+(y-1)22
Divide 1 by 1.
4y+5=9+3y-1+3y-1+(y-1)1
4y+5=9+3y-1+3y-1+(y-1)1
Simplify.
4y+5=9+3y-1+3y-1+y-1
4y+5=9+3y-1+3y-1+y-1
4y+5=9+3y-1+3y-1+y-1
Subtract 1 from 9.
4y+5=8+3y-1+3y-1+y
4y+5=8+6y-1+y
4y+5=8+6y-1+y
4y+5=8+6y-1+y
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
8+6y-1+y=4y+5
Move all terms not containing y-1 to the right side of the equation.
Subtract 8 from both sides of the equation.
6y-1+y=4y+5-8
Subtract y from both sides of the equation.
6y-1=4y+5-8-y
Subtract y from 4y.
6y-1=3y+5-8
Subtract 8 from 5.
6y-1=3y-3
6y-1=3y-3
Divide each term by 6 and simplify.
Divide each term in 6y-1=3y-3 by 6.
6y-16=3y6+-36
Cancel the common factor of 6.
Cancel the common factor.
6y-16=3y6+-36
Divide y-1 by 1.
y-1=3y6+-36
y-1=3y6+-36
Simplify each term.
Cancel the common factor of 3 and 6.
Factor 3 out of 3y.
y-1=3(y)6+-36
Cancel the common factors.
Factor 3 out of 6.
y-1=3y3⋅2+-36
Cancel the common factor.
y-1=3y3⋅2+-36
Rewrite the expression.
y-1=y2+-36
y-1=y2+-36
y-1=y2+-36
Cancel the common factor of -3 and 6.
Factor 3 out of -3.
y-1=y2+3(-1)6
Cancel the common factors.
Factor 3 out of 6.
y-1=y2+3⋅-13⋅2
Cancel the common factor.
y-1=y2+3⋅-13⋅2
Rewrite the expression.
y-1=y2+-12
y-1=y2+-12
y-1=y2+-12
Move the negative in front of the fraction.
y-1=y2-12
y-1=y2-12
y-1=y2-12
To remove the radical on the left side of the equation, square both sides of the equation.
y-12=(y2-12)2
Simplify each side of the equation.
Multiply the exponents in ((y-1)12)2.
Apply the power rule and multiply exponents, (am)n=amn.
(y-1)12⋅2=(y2-12)2
Cancel the common factor of 2.
Cancel the common factor.
(y-1)12⋅2=(y2-12)2
Rewrite the expression.
(y-1)1=(y2-12)2
(y-1)1=(y2-12)2
(y-1)1=(y2-12)2
Simplify.
y-1=(y2-12)2
y-1=(y2-12)2
Solve for y.
Simplify (y2-12)2.
Rewrite (y2-12)2 as (y2-12)(y2-12).
y-1=(y2-12)(y2-12)
Expand (y2-12)(y2-12) using the FOIL Method.
Apply the distributive property.
y-1=y2(y2-12)-12(y2-12)
Apply the distributive property.
y-1=y2⋅y2+y2(-12)-12(y2-12)
Apply the distributive property.
y-1=y2⋅y2+y2(-12)-12⋅y2-12(-12)
y-1=y2⋅y2+y2(-12)-12⋅y2-12(-12)
Simplify and combine like terms.
Simplify each term.
Multiply y2⋅y2.
Multiply y2 and y2.
y-1=y⋅y2⋅2+y2(-12)-12⋅y2-12(-12)
Raise y to the power of 1.
y-1=y1y2⋅2+y2(-12)-12⋅y2-12(-12)
Raise y to the power of 1.
y-1=y1y12⋅2+y2(-12)-12⋅y2-12(-12)
Use the power rule aman=am+n to combine exponents.
y-1=y1+12⋅2+y2(-12)-12⋅y2-12(-12)
y-1=y22⋅2+y2(-12)-12⋅y2-12(-12)
Multiply 2 by 2.
y-1=y24+y2(-12)-12⋅y2-12(-12)
y-1=y24+y2(-12)-12⋅y2-12(-12)
Multiply y2(-12).
Multiply y2 and 12.
y-1=y24-y2⋅2-12⋅y2-12(-12)
Multiply 2 by 2.
y-1=y24-y4-12⋅y2-12(-12)
y-1=y24-y4-12⋅y2-12(-12)
Multiply -12⋅y2.
Multiply y2 and 12.
y-1=y24-y4-y2⋅2-12(-12)
Multiply 2 by 2.
y-1=y24-y4-y4-12(-12)
y-1=y24-y4-y4-12(-12)
Multiply -12(-12).
Multiply -1 by -1.
y-1=y24-y4-y4+1(12)12
Multiply 12 by 1.
y-1=y24-y4-y4+12⋅12
Multiply 12 and 12.
y-1=y24-y4-y4+12⋅2
Multiply 2 by 2.
y-1=y24-y4-y4+14
y-1=y24-y4-y4+14
y-1=y24-y4-y4+14
Subtract y4 from -y4.
y-1=y24-2y4+14
y-1=y24-2y4+14
Simplify each term.
Cancel the common factor of 2.
Factor 2 out of -2.
y-1=y24+2(-1)y4+14
Factor 2 out of 4.
y-1=y24+2⋅-1y2⋅2+14
Cancel the common factor.
y-1=y24+2⋅-1y2⋅2+14
Rewrite the expression.
y-1=y24-1y2+14
y-1=y24-1y2+14
Rewrite -1y2 as -y2.
y-1=y24-y2+14
y-1=y24-y2+14
y-1=y24-y2+14
Since y is on the right side of the equation, switch the sides so it is on the left side of the equation.
y24-y2+14=y-1
Move all terms containing y to the left side of the equation.
Subtract y from both sides of the equation.
y24-y2+14-y=-1
To write -y as a fraction with a common denominator, multiply by 22.
y24-y2-y⋅22+14=-1
Combine -y and 22.
y24-y2+-y⋅22+14=-1
Combine the numerators over the common denominator.
y24+-y-y⋅22+14=-1
Find the common denominator.
Multiply -y-y⋅22 by 22.
y24+-y-y⋅22⋅22+14=-1
Combine.
y24+(-y-y⋅2)⋅22⋅2+14=-1
Multiply 2 by 2.
y24+(-y-y⋅2)⋅24+14=-1
y24+(-y-y⋅2)⋅24+14=-1
Combine the numerators over the common denominator.
y2+(-y-y⋅2)⋅2+14=-1
Simplify the numerator.
Multiply 2 by -1.
y2+(-y-2y)⋅2+14=-1
Subtract 2y from -y.
y2-3y⋅2+14=-1
Multiply 2 by -3.
y2-6y+14=-1
y2-6y+14=-1
y2-6y+14=-1
Multiply both sides of the equation by 4.
y2-6y+1=-1⋅4
Remove parentheses.
y2-6y+1=-1⋅4
Multiply -1 by 4.
y2-6y+1=-4
Move 4 to the left side of the equation by adding it to both sides.
y2-6y+1+4=0
y2-6y+5=0
Factor y2-6y+5 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 5 and whose sum is -6.
-5,-1
Write the factored form using these integers.
(y-5)(y-1)=0
(y-5)(y-1)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
y-5=0
y-1=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
y-5=0
Add 5 to both sides of the equation.
y=5
y=5
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
y-1=0
Add 1 to both sides of the equation.
y=1
y=1
The final solution is all the values that make (y-5)(y-1)=0 true.
y=5,1
y=5,1
y=5,1
Solve for y square root of 4y+5- square root of y-1=3

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