Solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth. Separate the solutions with a comma and list the smaller one first.

For the quadratic equation x^2 + 4 x = 4, the solution(s) is(are)

x^2 + 4 x = 4

x^2 + 4x - 4 = 0

Now just apply the formula.

To solve the quadratic equation x^2 + 4x = 4 using the quadratic formula, we first need to identify the values of a, b, and c. In this case, a = 1, b = 4, and c = -4.

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values of a, b, and c into the formula, we get:

x = (-4 ± √(4^2 - 4(1)(-4))) / (2(1))

Simplifying further:

x = (-4 ± √(16 + 16)) / 2

x = (-4 ± √(32)) / 2

Writing this equation with simplified values:

x = (-4 ± 4√2) / 2

Simplifying the expression further by canceling 2:

x = -2 ± 2√2

Therefore, the exact solutions to the quadratic equation x^2 + 4x = 4 are x = -2 + 2√2 and x = -2 - 2√2.

To determine the irrational solutions to the nearest thousandth using a calculator, we substitute these values into the calculator:

x ≈ -2 + 2√2 ≈ -0.171

x ≈ -2 - 2√2 ≈ -3.828

The irrational solutions to the quadratic equation x^2 + 4x = 4 to the nearest thousandth are approximately -0.171 and -3.828.