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

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

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