a^2+a=25

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Solution for a^2+a=25 equation:


Simplifying
a2 + a = 25

Reorder the terms:
a + a2 = 25

Solving
a + a2 = 25

Solving for variable 'a'.

Reorder the terms:
-25 + a + a2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + a + 25 + a2 = 0 + 25

Reorder the terms:
-25 + 25 + a + a2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + a + a2 = 0 + 25
a + a2 = 0 + 25

Combine like terms: 0 + 25 = 25
a + a2 = 25

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
a + 0.25 + a2 = 25 + 0.25

Reorder the terms:
0.25 + a + a2 = 25 + 0.25

Combine like terms: 25 + 0.25 = 25.25
0.25 + a + a2 = 25.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(a + 0.5) equal to 5.024937811 and -5.024937811.

Subproblem 1

a + 0.5 = 5.024937811 Simplifying a + 0.5 = 5.024937811 Reorder the terms: 0.5 + a = 5.024937811 Solving 0.5 + a = 5.024937811 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 5.024937811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 5.024937811 + -0.5 a = 5.024937811 + -0.5 Combine like terms: 5.024937811 + -0.5 = 4.524937811 a = 4.524937811 Simplifying a = 4.524937811

Subproblem 2

a + 0.5 = -5.024937811 Simplifying a + 0.5 = -5.024937811 Reorder the terms: 0.5 + a = -5.024937811 Solving 0.5 + a = -5.024937811 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -5.024937811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -5.024937811 + -0.5 a = -5.024937811 + -0.5 Combine like terms: -5.024937811 + -0.5 = -5.524937811 a = -5.524937811 Simplifying a = -5.524937811

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.524937811, -5.524937811}

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