(x-3)(x-7)=4

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Solution for (x-3)(x-7)=4 equation:


Simplifying
(x + -3)(x + -7) = 4

Reorder the terms:
(-3 + x)(x + -7) = 4

Reorder the terms:
(-3 + x)(-7 + x) = 4

Multiply (-3 + x) * (-7 + x)
(-3(-7 + x) + x(-7 + x)) = 4
((-7 * -3 + x * -3) + x(-7 + x)) = 4
((21 + -3x) + x(-7 + x)) = 4
(21 + -3x + (-7 * x + x * x)) = 4
(21 + -3x + (-7x + x2)) = 4

Combine like terms: -3x + -7x = -10x
(21 + -10x + x2) = 4

Solving
21 + -10x + x2 = 4

Solving for variable 'x'.

Reorder the terms:
21 + -4 + -10x + x2 = 4 + -4

Combine like terms: 21 + -4 = 17
17 + -10x + x2 = 4 + -4

Combine like terms: 4 + -4 = 0
17 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-17' to each side of the equation.
17 + -10x + -17 + x2 = 0 + -17

Reorder the terms:
17 + -17 + -10x + x2 = 0 + -17

Combine like terms: 17 + -17 = 0
0 + -10x + x2 = 0 + -17
-10x + x2 = 0 + -17

Combine like terms: 0 + -17 = -17
-10x + x2 = -17

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

Add '25' to each side of the equation.
-10x + 25 + x2 = -17 + 25

Reorder the terms:
25 + -10x + x2 = -17 + 25

Combine like terms: -17 + 25 = 8
25 + -10x + x2 = 8

Factor a perfect square on the left side:
(x + -5)(x + -5) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + -5) equal to 2.828427125 and -2.828427125.

Subproblem 1

x + -5 = 2.828427125 Simplifying x + -5 = 2.828427125 Reorder the terms: -5 + x = 2.828427125 Solving -5 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 2.828427125 + 5 Combine like terms: -5 + 5 = 0 0 + x = 2.828427125 + 5 x = 2.828427125 + 5 Combine like terms: 2.828427125 + 5 = 7.828427125 x = 7.828427125 Simplifying x = 7.828427125

Subproblem 2

x + -5 = -2.828427125 Simplifying x + -5 = -2.828427125 Reorder the terms: -5 + x = -2.828427125 Solving -5 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -2.828427125 + 5 Combine like terms: -5 + 5 = 0 0 + x = -2.828427125 + 5 x = -2.828427125 + 5 Combine like terms: -2.828427125 + 5 = 2.171572875 x = 2.171572875 Simplifying x = 2.171572875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.828427125, 2.171572875}

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