How To Complete The Square: Steps For Solving Quadratic Equations And Finding Vertex

Steps to Complete the Square

1. Move Constant to the right2. Add (coefficient/2)^2 to both sides3. Square both sides4. Split into 2 possible solutions5. Get 2 solutions

Completing the square is a useful algebraic technique that is used to transform a quadratic equation into a perfect square trinomial. This technique can then be used to solve the quadratic equation, find the vertex of a parabolic graph, and solve for minima or maxima of certain functions.

The following are the steps to complete the square:

Step 1: Write the quadratic equation in the standard form: ax^2 + bx + c = 0.

Step 2: Divide the entire equation by the coefficient of the x^2 term, a, to obtain a reduced form if necessary.

Step 3: Move the constant term, c, to the other side of the equation by subtracting it from both sides.

Step 4: Add and subtract the square of half the coefficient of the x term (b/2a) to create a perfect square trinomial. This is achieved by adding (b/2a)^2 to both sides of the equation:

ax^2 + bx = -c —> ax^2 + bx + (b/2a)^2 = (b/2a)^2 – c

Step 5: Factor the perfect square trinomial obtained in step 4, and simplify the right-hand side by finding a common denominator if necessary:

(ax + b/2a)^2 = (4ac – b^2)/4a^2

Step 6: Take the square root of both sides of the equation to solve for x, remembering to account for the positive and negative square roots.

ax + b/2a = ±√(4ac – b^2)/2a —> x = (-b ± √(b^2 – 4ac))/2a

These six steps make up the process of completing the square. With this technique, we can transform almost any quadratic equation into a perfect square trinomial and then solve for the roots of the equation.

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