Solve by completing the squareSolve by completing the square could take a little bit more time to do than solving by factoring. However, the steps are straightforward. Before showing examples, you need to understand what a perfect square trinomial is Binomial × same Binomial = Perfect square trinomial (x + 4) × (x + 4) = x^{2} + 4x + 4x + 16 = x^{2} + 8x + 16 (x + a) × (x + a) = x^{2} + ax + ax + a^{2} = x^{2} + 2ax + a^{2} Important observation: What is the relationship between the coefficient of the second term and the last term? For x^{2} + 8x + 16, coefficient of the second term is 8 and the last term is 16 (8/2)^{2} = 4^{2} = 16 For, x^{2} + 2ax + a^{2}, the coefficient of the second term is 2a and the last term is a^{2} (2a/2)^{2} = a^{2} So, what is the relationship? The last term is obtained by dividing the coefficient of the second term by 2 and squaring the result Now, let's say you have x^{2} + 20x and you want to find the last term to make the whole thing a perfect square trinomial Just do (20/2)^{2} = 10^{2}. The perfect square trinomial is x^{2} + 20x + 10^{2}= (x + 10) × (x + 10) Example #1: Solve by completing the square x^{2} + 6x + 8 = 0 x^{2} + 6x + 8 = 0 Subtract 8 from both sides of the equation. x^{2} + 6x + 8  8 = 0  8 x^{2} + 6x =  8 To complete the square, always do the following 24 hours a day 365 days a year. It is never going to change when you solve by completing the square! You are basically looking for a term to add to x^{2} + 6x that will make it a perfect square trinomial. To this end, get the coefficient of the second term, divide it by 2 and raise it to the second power. The second term is 6x and the coefficient is 6. 6/2 = 3 and after squaring 3, we get 3^{2} x^{2} + 6x =  8 Add 3^{2} to both sides of the equation above x^{2} + 6x + 3^{2} =  8 + 3^{2} x^{2} + 6x + 3^{2} =  8 + 3^{2} (x + 3)^{2} = 8 + 9 (x + 3)^{2} = 1 Take the square root of both sides √((x + 3)^{2}) = √(1) x + 3 = ±1 When x + 3 = 1, x = 2 When x + 3 = 1, x = 4 Example #2: Solve by completing the square x^{2} + 6x + 8 = 0 instead of x^{2} + 6x + 8 = 0 The second term this time is 6x and the coefficient is 6. 6/2 = 3 and after squaring 3, we get (3)^{2} = 9 x^{2} + 6x =  8 Add (3)^{2} to both sides of the equation above x^{2} + 6x + (3)^{2} =  8 + (3)^{2} (x + 3)^{2} = 8 + 9 (x + 3)^{2} = 1 Take the square root of both sides √((x + 3)^{2}) = √(1) x + 3 = ±1 When x + 3 = 1, x = 4 When x + 3 = 1, x = 2 Example #3: Solve by completing the square 3x^{2} + 8x + 3 = 0 3x^{2} + 8x + 3 = 0 Divide everything by 3. Always do that when the coefficient of the first term is not 1 (3/3)x^{2}+ (8/3)x + 3/3 = 0/3 x^{2}+ (8/3)x + 1 = 0 Add 1 to both sides of the equation. x^{2} + (8/3)x + 1 + 1 = 0 + 1 x^{2} + (8/3)x = 1 The second term is (8/3)x and the coefficient is 8/3. 8/3 ÷ 2 = 8/3 × 1/2 = 8/6 and after squaring 8/6, we get (8/6)^{2} x^{2} + (8/3)x = 1 Add (8/6)^{2} to both sides of the equation above x^{2} + (8/3)x + (8/6)^{2} = 1 + (8/6)^{2} (x + 8/6)^{2} = 1 + 64/36 (x + 8/6)^{2} = 36/36 + 64/36 = (36 + 64)/36 = 100/36 Take the square root of both sides √((x + 8/6)^{2}) = √(100/36) x + 8/6 = ±10/6 x + 8/6 = 10/6 x = 10/6  8/6 = 2/6 = 1/3 x + 8/6 =  10/6 x = 10/6  8/6 = 18/6 = 3 To solve by completing the square can become quickly hard as shwon in example #3 




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