# Solve for s (3+4s)/2=2 square root of 1+s

3+4s2=21+s
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
21+s=3+4s2
To remove the radical on the left side of the equation, square both sides of the equation.
(21+s)2=(3+4s2)2
Simplify each side of the equation.
Apply the product rule to 2(1+s)12.
22((1+s)12)2=(3+4s2)2
Raise 2 to the power of 2.
4((1+s)12)2=(3+4s2)2
Multiply the exponents in ((1+s)12)2.
Apply the power rule and multiply exponents, (am)n=amn.
4(1+s)12⋅2=(3+4s2)2
Cancel the common factor of 2.
Cancel the common factor.
4(1+s)12⋅2=(3+4s2)2
Rewrite the expression.
4(1+s)1=(3+4s2)2
4(1+s)1=(3+4s2)2
4(1+s)1=(3+4s2)2
Simplify.
4(1+s)=(3+4s2)2
Apply the distributive property.
4⋅1+4s=(3+4s2)2
Multiply 4 by 1.
4+4s=(3+4s2)2
Apply the product rule to 3+4s2.
4+4s=(3+4s)222
Raise 2 to the power of 2.
4+4s=(3+4s)24
4+4s=(3+4s)24
Solve for s.
Move all terms containing s to the left side of the equation.
Subtract (3+4s)24 from both sides of the equation.
4+4s-(3+4s)24=0
To write 4 as a fraction with a common denominator, multiply by 44.
4s+4⋅44-(3+4s)24=0
Combine 4 and 44.
4s+4⋅44-(3+4s)24=0
Combine the numerators over the common denominator.
4s+4⋅4-(3+4s)24=0
Multiply 4 by 4.
4s+16-(3+4s)24=0
Simplify the numerator.
Rewrite 16 as 42.
4s+42-(3+4s)24=0
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=4 and b=3+4s.
4s+(4+3+4s)(4-(3+4s))4=0
Simplify.
Add 4 and 3.
4s+(7+4s)(4-(3+4s))4=0
Apply the distributive property.
4s+(7+4s)(4-1⋅3-(4s))4=0
Multiply -1 by 3.
4s+(7+4s)(4-3-(4s))4=0
Multiply 4 by -1.
4s+(7+4s)(4-3-4s)4=0
Subtract 3 from 4.
4s+(7+4s)(1-4s)4=0
4s+(7+4s)(1-4s)4=0
4s+(7+4s)(1-4s)4=0
To write 4s as a fraction with a common denominator, multiply by 44.
4s⋅44+(7+4s)(1-4s)4=0
Combine 4s and 44.
4s⋅44+(7+4s)(1-4s)4=0
Combine the numerators over the common denominator.
4s⋅4+(7+4s)(1-4s)4=0
Simplify the numerator.
Multiply 4 by 4.
16s+(7+4s)(1-4s)4=0
Expand (7+4s)(1-4s) using the FOIL Method.
Apply the distributive property.
16s+7(1-4s)+4s(1-4s)4=0
Apply the distributive property.
16s+7⋅1+7(-4s)+4s(1-4s)4=0
Apply the distributive property.
16s+7⋅1+7(-4s)+4s⋅1+4s(-4s)4=0
16s+7⋅1+7(-4s)+4s⋅1+4s(-4s)4=0
Simplify and combine like terms.
Simplify each term.
Multiply 7 by 1.
16s+7+7(-4s)+4s⋅1+4s(-4s)4=0
Multiply -4 by 7.
16s+7-28s+4s⋅1+4s(-4s)4=0
Multiply 4 by 1.
16s+7-28s+4s+4s(-4s)4=0
Multiply s by s.
16s+7-28s+4s+4⋅-4s24=0
Multiply 4 by -4.
16s+7-28s+4s-16s24=0
16s+7-28s+4s-16s24=0
Add -28s and 4s.
16s+7-24s-16s24=0
16s+7-24s-16s24=0
Subtract 24s from 16s.
-8s+7-16s24=0
Reorder terms.
-16s2-8s+74=0
-16s2-8s+74=0
Factor -1 out of -16s2.
-(16s2)-8s+74=0
Factor -1 out of -8s.
-(16s2)-(8s)+74=0
Factor -1 out of -(16s2)-(8s).
-(16s2+8s)+74=0
Rewrite 7 as -1(-7).
-(16s2+8s)-1(-7)4=0
Factor -1 out of -(16s2+8s)-1(-7).
-(16s2+8s-7)4=0
Rewrite -(16s2+8s-7) as -1(16s2+8s-7).
-1(16s2+8s-7)4=0
Move the negative in front of the fraction.
-16s2+8s-74=0
-16s2+8s-74=0
Multiply both sides of the equation by -4.
-4⋅(-16s2+8s-74)=-4⋅0
Simplify both sides of the equation.
Simplify -4⋅(-16s2+8s-74).
Cancel the common factor of 4.
Move the leading negative in -16s2+8s-74 into the numerator.
-4⋅-(16s2+8s-7)4=-4⋅0
Factor 4 out of -4.
4(-1)⋅-(16s2+8s-7)4=-4⋅0
Cancel the common factor.
4⋅-1⋅-(16s2+8s-7)4=-4⋅0
Rewrite the expression.
-1⋅(-(16s2+8s-7))=-4⋅0
-1⋅(-(16s2+8s-7))=-4⋅0
Multiply.
Multiply -1 by -1.
1⋅(16s2+8s-7)=-4⋅0
Multiply 16s2+8s-7 by 1.
16s2+8s-7=-4⋅0
16s2+8s-7=-4⋅0
16s2+8s-7=-4⋅0
Multiply -4 by 0.
16s2+8s-7=0
16s2+8s-7=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=16, b=8, and c=-7 into the quadratic formula and solve for s.
-8±82-4⋅(16⋅-7)2⋅16
Simplify.
Simplify the numerator.
Raise 8 to the power of 2.
s=-8±64-4⋅(16⋅-7)2⋅16
Multiply 16 by -7.
s=-8±64-4⋅-1122⋅16
Multiply -4 by -112.
s=-8±64+4482⋅16
Add 64 and 448.
s=-8±5122⋅16
Rewrite 512 as 162⋅2.
Factor 256 out of 512.
s=-8±256(2)2⋅16
Rewrite 256 as 162.
s=-8±162⋅22⋅16
s=-8±162⋅22⋅16
Pull terms out from under the radical.
s=-8±1622⋅16
s=-8±1622⋅16
Multiply 2 by 16.
s=-8±16232
Simplify -8±16232.
s=-1±224
s=-1±224
The final answer is the combination of both solutions.
s=-1-224,-1+224
s=-1-224,-1+224
Exclude the solutions that do not make 3+4s2=21+s true.
s=-1-224
The result can be shown in multiple forms.
Exact Form:
s=-1-224
Decimal Form:
s=0.45710678…
Solve for s (3+4s)/2=2 square root of 1+s

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