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

Multiply the exponents in ((4y2-3)14)4.

Apply the power rule and multiply exponents, (am)n=amn.

(4y2-3)14⋅4=(-y)4

Cancel the common factor of 4.

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

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.

Factor -1 out of -u2+4u-3.

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.

Factor u2-4u+3 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 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

Multiply each term in -(u-3)(u-1)=0 by -1.

(-(u-3)(u-1))⋅-1=0⋅-1

Simplify (-(u-3)(u-1))⋅-1.

Simplify by multiplying through.

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.

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.

Simplify each term.

Multiply u by u by adding the exponents.

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.

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.

Multiply -u2⋅-1.

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.

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.

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.

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.

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.

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.

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.

Simplify the right side of the equation.

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.

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