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.