Solve for d d^2 = square root of d+2

d2=d+2
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
d+2=d2
To remove the radical on the left side of the equation, square both sides of the equation.
d+22=(d2)2
Simplify each side of the equation.
Multiply the exponents in ((d+2)12)2.
Apply the power rule and multiply exponents, (am)n=amn.
(d+2)12⋅2=(d2)2
Cancel the common factor of 2.
Cancel the common factor.
(d+2)12⋅2=(d2)2
Rewrite the expression.
(d+2)1=(d2)2
(d+2)1=(d2)2
(d+2)1=(d2)2
Simplify.
d+2=(d2)2
Multiply the exponents in (d2)2.
Apply the power rule and multiply exponents, (am)n=amn.
d+2=d2⋅2
Multiply 2 by 2.
d+2=d4
d+2=d4
d+2=d4
Solve for d.
Subtract d4 from both sides of the equation.
d+2-d4=0
Factor -1 out of d+2-d4.
Reorder the expression.
Move 2.
d-d4+2=0
Reorder d and -d4.
-d4+d+2=0
-d4+d+2=0
Factor -1 out of -d4.
-(d4)+d+2=0
Factor -1 out of d.
-(d4)-1(-d)+2=0
Rewrite 2 as -1(-2).
-(d4)-1(-d)-1⋅-2=0
Factor -1 out of -(d4)-1(-d).
-(d4-d)-1⋅-2=0
Factor -1 out of -(d4-d)-1(-2).
-(d4-d-2)=0
-(d4-d-2)=0
Multiply each term in -(d4-d-2)=0 by -1
Multiply each term in -(d4-d-2)=0 by -1.
-(d4-d-2)⋅-1=0⋅-1
Simplify -(d4-d-2)⋅-1.
Apply the distributive property.
(-d4–d–2)⋅-1=0⋅-1
Simplify.
Multiply –d.
Multiply -1 by -1.
(-d4+1d–2)⋅-1=0⋅-1
Multiply d by 1.
(-d4+d–2)⋅-1=0⋅-1
(-d4+d–2)⋅-1=0⋅-1
Multiply -1 by -2.
(-d4+d+2)⋅-1=0⋅-1
(-d4+d+2)⋅-1=0⋅-1
Apply the distributive property.
-d4⋅-1+d⋅-1+2⋅-1=0⋅-1
Simplify.
Multiply -d4⋅-1.
Multiply -1 by -1.
1d4+d⋅-1+2⋅-1=0⋅-1
Multiply d4 by 1.
d4+d⋅-1+2⋅-1=0⋅-1
d4+d⋅-1+2⋅-1=0⋅-1
Move -1 to the left of d.
d4-1⋅d+2⋅-1=0⋅-1
Multiply 2 by -1.
d4-1⋅d-2=0⋅-1
d4-1⋅d-2=0⋅-1
Rewrite -1d as -d.
d4-d-2=0⋅-1
d4-d-2=0⋅-1
Multiply 0 by -1.
d4-d-2=0
d4-d-2=0
If a polynomial function has integer coefficients, then every rational zero will have the form pq where p is a factor of the constant and q is a factor of the leading coefficient.
p=±1,±2
q=±1
Find every combination of ±pq. These are the possible roots of the polynomial function.
±1,±2
Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is 0, which means it is a root.
(-1)4-(-1)-2
Simplify the expression. In this case, the expression is equal to 0 so d=-1 is a root of the polynomial.
Simplify each term.
Raise -1 to the power of 4.
1-(-1)-2
Multiply -1 by -1.
1+1-2
1+1-2
2-2
Subtract 2 from 2.
0
0
0
Since -1 is a known root, divide the polynomial by d+1 to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
d4-d-2d+1
Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by 1.
Place the numbers representing the divisor and the dividend into a division-like configuration.
 -1 1 0 0 -1 -2
The first number in the dividend (1) is put into the first position of the result area (below the horizontal line).
 -1 1 0 0 -1 -2 1
Multiply the newest entry in the result (1) by the divisor (-1) and place the result of (-1) under the next term in the dividend (0).
 -1 1 0 0 -1 -2 -1 1
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 -1 1 0 0 -1 -2 -1 1 -1
Multiply the newest entry in the result (-1) by the divisor (-1) and place the result of (1) under the next term in the dividend (0).
 -1 1 0 0 -1 -2 -1 1 1 -1
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 -1 1 0 0 -1 -2 -1 1 1 -1 1
Multiply the newest entry in the result (1) by the divisor (-1) and place the result of (-1) under the next term in the dividend (-1).
 -1 1 0 0 -1 -2 -1 1 -1 1 -1 1
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 -1 1 0 0 -1 -2 -1 1 -1 1 -1 1 -2
Multiply the newest entry in the result (-2) by the divisor (-1) and place the result of (2) under the next term in the dividend (-2).
 -1 1 0 0 -1 -2 -1 1 -1 2 1 -1 1 -2
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 -1 1 0 0 -1 -2 -1 1 -1 2 1 -1 1 -2 0
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
1d3+-1d2+(1)d-2
Simplify the quotient polynomial.
d3-d2+d-2
d3-d2+d-2
Graph each side of the equation. The solution is the x-value of the point of intersection.
d≈1.35320996
The polynomial can be written as a set of linear factors.
(d+1)(d-1.35320996)
These are the roots (zeros) of the polynomial d4-d-2.
d=-1,1.35320996
d=-1,1.35320996
Solve for d d^2 = square root of d+2

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