To solve the quadratic equation x² + 4x + 7 = 0 by completing the square, we start by rearranging the equation. Our goal is to express the quadratic in the form of a perfect square.
First, we need the equation in the standard form:
x² + 4x = -7
Next, we complete the square for the left side. We take the coefficient of x, which is 4, divide it by 2 to get 2, and then square it to get 4.
Now, we add and subtract this square (4) to the left side:
x² + 4x + 4 – 4 = -7
This simplifies to:
(x + 2)² – 4 = -7
Next, we add 4 to both sides to isolate the perfect square:
(x + 2)² = -3
Now we notice that we have a negative number on the right side, which means the solutions will involve imaginary numbers. We take the square root of both sides:
x + 2 = ±√(-3)
We can simplify √(-3) to √3 * i (where i is the imaginary unit). Thus, we have:
x + 2 = ±√3 * i
Finally, we subtract 2 from both sides to find x:
x = -2 ± √3 * i
So the two solutions to the quadratic equation are:
x = -2 + √3 * i and x = -2 – √3 * i
These solutions indicate that the parabola represented by the equation does not intersect the x-axis, confirming that the solutions are complex numbers.