t^2+10t=-22

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Solution for t^2+10t=-22 equation:


Simplifying
t2 + 10t = -22

Reorder the terms:
10t + t2 = -22

Solving
10t + t2 = -22

Solving for variable 't'.

Reorder the terms:
22 + 10t + t2 = -22 + 22

Combine like terms: -22 + 22 = 0
22 + 10t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + 10t + -22 + t2 = 0 + -22

Reorder the terms:
22 + -22 + 10t + t2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + 10t + t2 = 0 + -22
10t + t2 = 0 + -22

Combine like terms: 0 + -22 = -22
10t + t2 = -22

The t term is 10t.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10t + 25 + t2 = -22 + 25

Reorder the terms:
25 + 10t + t2 = -22 + 25

Combine like terms: -22 + 25 = 3
25 + 10t + t2 = 3

Factor a perfect square on the left side:
(t + 5)(t + 5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(t + 5) equal to 1.732050808 and -1.732050808.

Subproblem 1

t + 5 = 1.732050808 Simplifying t + 5 = 1.732050808 Reorder the terms: 5 + t = 1.732050808 Solving 5 + t = 1.732050808 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + t = 1.732050808 + -5 Combine like terms: 5 + -5 = 0 0 + t = 1.732050808 + -5 t = 1.732050808 + -5 Combine like terms: 1.732050808 + -5 = -3.267949192 t = -3.267949192 Simplifying t = -3.267949192

Subproblem 2

t + 5 = -1.732050808 Simplifying t + 5 = -1.732050808 Reorder the terms: 5 + t = -1.732050808 Solving 5 + t = -1.732050808 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + t = -1.732050808 + -5 Combine like terms: 5 + -5 = 0 0 + t = -1.732050808 + -5 t = -1.732050808 + -5 Combine like terms: -1.732050808 + -5 = -6.732050808 t = -6.732050808 Simplifying t = -6.732050808

Solution

The solution to the problem is based on the solutions from the subproblems. t = {-3.267949192, -6.732050808}

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