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

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

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

Simplify by adding and subtracting.

Add 1 and 1.

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 |

-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