Completing the Square Completing the Square Some quadratics are fairly simple to solve because they are of the form "something-with-x squared equals some number", and then you take the square root of both sides. An example would be: (x – 4)2 = 5 x – 4 = ± sqrt(5) x = 4 ± sqrt(5) x = 4 – sqrt(5) and x = 4 + sqrt(5) Unfortunately, most quadratics don't come neatly squared like this. For your average everyday quadratic, you first have to use the technique of "completing the square" to rearrange the quadratic into the neat "(squared part) equals (a number)" format demonstrated above. For example:
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We are very much aware of formula for Square of addition or subtraction of two Numbers which can be given as: (a + b) 2 = a2 + b2 + 2ab and (a - b) 2 = a2 + b2 – 2ab, When you have a Quadratic Equation of form ax² + bx + c which is not possible to be factorized, you can use technique called completing the square. To complete the square means creating a polynomial with three terms that results into a perfect square. To start with, rewrite the quadratic expression given as ax² + bx + c and move the constant term 'c' to right side of equation to get the form ax² + bx = - c. Divide this complete equation constant factor “a” if a≠ 1 to get x² + (b / a) x = -c / a. Divide coefficient of 'x' i.e. (b / a) by 2 and it now becomes (b / 2a) and then square it to get (b / 2a) ². Add (b / 2a)² to both sides of equation to get: x² + (b / a) x + (b / 2a)² = -c / a + (b / 2a) ². Next step would we writing left side of equation as perfect square: [x + (b / 2a)] ² = -c / a + (b / 2a) ². For example, let’s take an expression: 4x² + 16x - 20. Where, a = 4, b = 16 and c = -20.
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Moving constant 'c' to right side we get 4x² + 16x = 20. Next divide both sides of equation by 4 to get: x² + 4x = 20 / 4. Taking half of 4 which is coefficient of 'x' and then squaring it to get: (4 / 2) ² = 4, Add 4 to your equation: x² + 4x + 4 = 5 + 4, or x² + 4x + 4 = 9, Making left side a perfect square we get: (x + 2) 2 = 9, or x = 1, -5.
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