Completing the Square Calculator

Completing the square is an algebraic method that transforms a quadratic expression ax^2 + bx + c into vertex form a(x - h)^2 + k. This is done by adding and subtracting a specific value (b/2a)^2 to create a perfect square trinomial. The resulting form directly reveals the vertex of the parabola and makes it easy to solve or graph the equation.

Step 1: Start with ax^2 + bx + c. Step 2: If a is not 1, factor it from the first two terms. Step 3: Take half the coefficient of x (that is b/2a). Step 4: Square this value to get (b/2a)^2. Step 5: Add and subtract this inside the parentheses. Step 6: Rewrite as a perfect square plus a constant: a(x + b/2a)^2 + (c - b^2/4a).

Completing the square has many uses: (1) Finding the vertex of a parabola directly, (2) Solving quadratic equations without the quadratic formula, (3) Deriving the quadratic formula itself, (4) Converting circle equations to standard form (x-h)^2 + (y-k)^2 = r^2, (5) Simplifying integration problems in calculus, and (6) Understanding the geometric transformations of a parabola.

Standard form is ax^2 + bx + c where a, b, c are constants. Vertex form is a(x - h)^2 + k where (h, k) is the vertex. Standard form is easier for finding y-intercept (just c) and for polynomial operations. Vertex form is better for graphing since you can immediately see the vertex and the direction the parabola opens.

After completing the square, the equation is in the form a(x - h)^2 + k. The vertex is simply the point (h, k). For example, if you get 2(x - 3)^2 + 5, the vertex is (3, 5). Remember: the h value has the opposite sign of what appears in the parentheses (x - 3 means h = 3, not -3).

If a is not 1, factor it out from the x^2 and x terms first: ax^2 + bx + c = a(x^2 + (b/a)x) + c. Then complete the square inside the parentheses using (b/2a)^2. Remember that when you add (b/2a)^2 inside the parentheses, you are actually adding a times (b/2a)^2, so you must subtract a times (b/2a)^2 = b^2/4a outside.

Yes, completing the square can solve any quadratic equation, including those with complex solutions. After converting to vertex form a(x - h)^2 + k = 0, solve for x: (x - h)^2 = -k/a, then x = h plus or minus sqrt(-k/a). If -k/a is negative, the solutions involve imaginary numbers. This method is actually how the quadratic formula is derived.

The quadratic formula x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a is derived by completing the square on the general equation ax^2 + bx + c = 0. Working through the steps algebraically with variables instead of numbers produces the formula. This is why the -b/2a term appears in both methods - it is the x-coordinate of the vertex.