Completing the Square Calculator

The method of completing the square transforms a quadratic expression from standard form, ax² + bx + c = 0, into the vertex or binomial squared form, a(x - h)² + k = 0. First, the leading coefficient 'a' is factored out or divided across the entire equation, yielding b1 = b/a and c1 = c/a. Next, we take the linear coefficient term, divide it by two to get b2 = b1/2, and square that value to find b2_sq. By adding and balancing this value within the expression, the polynomial neatly collapses into a binomial square, enabling precise calculation of both real and complex roots (x1 and x2) using square roots and arithmetic adjustments.

Let us walk through completing the square for the quadratic equation 2x² + 8x + 6 = 0. Here, a = 2, b = 8, and c = 6. First, divide all terms by a (2) to simplify the equation, resulting in x² + 4x + 3 = 0, meaning b1 = 4 and c1 = 3. Next, take half of the linear term b1 (4 / 2 = 2) and square it to get 4. We rewrite the equation by incorporating this value: (x + 2)² - 4 + 3 = 0, which simplifies to (x + 2)² - 1 = 0. Moving the constant to the right side gives (x + 2)² = 1. Taking the square root of both sides yields x + 2 = ±1, resulting in our final roots: x = -1 and x = -3.

Always ensure your quadratic equation is arranged in standard descending order (ax² + bx + c = 0) before plugging in your variables. If your leading coefficient 'a' is negative, be extra careful with sign distributions during the division and square root steps. Utilizing a precision setting helps manage irrational numbers and long decimals when dealing with non-integer roots.