Solve for y fourth root of 4y^2-3=-y

Math
4y2-34=-y
To remove the radical on the left side of the equation, raise both sides of the equation to the power of 4.
4y2-344=(-y)4
Simplify each side of the equation.
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Multiply the exponents in ((4y2-3)14)4.
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Apply the power rule and multiply exponents, (am)n=amn.
(4y2-3)14⋅4=(-y)4
Cancel the common factor of 4.
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Cancel the common factor.
(4y2-3)14⋅4=(-y)4
Rewrite the expression.
(4y2-3)1=(-y)4
(4y2-3)1=(-y)4
(4y2-3)1=(-y)4
Simplify.
4y2-3=(-y)4
Apply the product rule to -y.
4y2-3=(-1)4y4
Raise -1 to the power of 4.
4y2-3=1y4
Multiply y4 by 1.
4y2-3=y4
4y2-3=y4
Solve for y.
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Subtract y4 from both sides of the equation.
4y2-3-y4=0
Substitute u=y2 into the equation. This will make the quadratic formula easy to use.
-u2+4u-3=0
u=y2
Factor the left side of the equation.
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Factor -1 out of -u2+4u-3.
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Factor -1 out of -u2.
-(u2)+4u-3=0
Factor -1 out of 4u.
-(u2)-(-4u)-3=0
Rewrite -3 as -1(3).
-(u2)-(-4u)-1⋅3=0
Factor -1 out of -(u2)-(-4u).
-(u2-4u)-1⋅3=0
Factor -1 out of -(u2-4u)-1(3).
-(u2-4u+3)=0
-(u2-4u+3)=0
Factor.
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Factor u2-4u+3 using the AC method.
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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 3 and whose sum is -4.
-3,-1
Write the factored form using these integers.
-((u-3)(u-1))=0
-((u-3)(u-1))=0
Remove unnecessary parentheses.
-(u-3)(u-1)=0
-(u-3)(u-1)=0
-(u-3)(u-1)=0
Multiply each term in -(u-3)(u-1)=0 by -1
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Multiply each term in -(u-3)(u-1)=0 by -1.
(-(u-3)(u-1))⋅-1=0⋅-1
Simplify (-(u-3)(u-1))⋅-1.
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Simplify by multiplying through.
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Apply the distributive property.
(-u–3)(u-1)⋅-1=0⋅-1
Multiply -1 by -3.
(-u+3)(u-1)⋅-1=0⋅-1
(-u+3)(u-1)⋅-1=0⋅-1
Expand (-u+3)(u-1) using the FOIL Method.
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Apply the distributive property.
(-u(u-1)+3(u-1))⋅-1=0⋅-1
Apply the distributive property.
(-u⋅u-u⋅-1+3(u-1))⋅-1=0⋅-1
Apply the distributive property.
(-u⋅u-u⋅-1+3u+3⋅-1)⋅-1=0⋅-1
(-u⋅u-u⋅-1+3u+3⋅-1)⋅-1=0⋅-1
Simplify and combine like terms.
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Simplify each term.
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Multiply u by u by adding the exponents.
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Move u.
(-(u⋅u)-u⋅-1+3u+3⋅-1)⋅-1=0⋅-1
Multiply u by u.
(-u2-u⋅-1+3u+3⋅-1)⋅-1=0⋅-1
(-u2-u⋅-1+3u+3⋅-1)⋅-1=0⋅-1
Multiply -u⋅-1.
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Multiply -1 by -1.
(-u2+1u+3u+3⋅-1)⋅-1=0⋅-1
Multiply u by 1.
(-u2+u+3u+3⋅-1)⋅-1=0⋅-1
(-u2+u+3u+3⋅-1)⋅-1=0⋅-1
Multiply 3 by -1.
(-u2+u+3u-3)⋅-1=0⋅-1
(-u2+u+3u-3)⋅-1=0⋅-1
Add u and 3u.
(-u2+4u-3)⋅-1=0⋅-1
(-u2+4u-3)⋅-1=0⋅-1
Apply the distributive property.
-u2⋅-1+4u⋅-1-3⋅-1=0⋅-1
Simplify.
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Multiply -u2⋅-1.
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Multiply -1 by -1.
1u2+4u⋅-1-3⋅-1=0⋅-1
Multiply u2 by 1.
u2+4u⋅-1-3⋅-1=0⋅-1
u2+4u⋅-1-3⋅-1=0⋅-1
Multiply -1 by 4.
u2-4u-3⋅-1=0⋅-1
Multiply -3 by -1.
u2-4u+3=0⋅-1
u2-4u+3=0⋅-1
u2-4u+3=0⋅-1
Multiply 0 by -1.
u2-4u+3=0
u2-4u+3=0
Factor u2-4u+3 using the AC method.
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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 3 and whose sum is -4.
-3,-1
Write the factored form using these integers.
(u-3)(u-1)=0
(u-3)(u-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.
u-3=0
u-1=0
Set the first factor equal to 0 and solve.
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Set the first factor equal to 0.
u-3=0
Add 3 to both sides of the equation.
u=3
u=3
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
u-1=0
Add 1 to both sides of the equation.
u=1
u=1
The final solution is all the values that make (u-3)(u-1)=0 true.
u=3,1
Substitute the real value of u=y2 back into the solved equation.
y2=3
(y2)1=1
Solve the first equation for y.
y2=3
Solve the equation for y.
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Take the square root of both sides of the equation to eliminate the exponent on the left side.
y=±3
The complete solution is the result of both the positive and negative portions of the solution.
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First, use the positive value of the ± to find the first solution.
y=3
Next, use the negative value of the ± to find the second solution.
y=-3
The complete solution is the result of both the positive and negative portions of the solution.
y=3,-3
y=3,-3
y=3,-3
Solve the second equation for y.
(y2)1=1
Solve the equation for y.
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Take the 1th root of each side of the equation to set up the solution for y
(y2)1⋅11=11
Remove the perfect root factor y2 under the radical to solve for y.
y2=11
Take the square root of both sides of the equation to eliminate the exponent on the left side.
y=±11
The complete solution is the result of both the positive and negative portions of the solution.
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Simplify the right side of the equation.
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Any root of 1 is 1.
y=±1
Any root of 1 is 1.
y=±1
y=±1
The complete solution is the result of both the positive and negative portions of the solution.
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First, use the positive value of the ± to find the first solution.
y=1
Next, use the negative value of the ± to find the second solution.
y=-1
The complete solution is the result of both the positive and negative portions of the solution.
y=1,-1
y=1,-1
y=1,-1
y=1,-1
The solution to -y4+4y2-3=0 is y=3,-3,1,-1.
y=3,-3,1,-1
y=3,-3,1,-1
Exclude the solutions that do not make 4y2-34=-y true.
y=-3,-1
The result can be shown in multiple forms.
Exact Form:
y=-3,-1
Decimal Form:
y=-1.73205080…,-1
Solve for y fourth root of 4y^2-3=-y

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