Solve 2x2 - 20x + 49 = 0 by completing the square.
To solve a problem by completing the square, I recommend first dividing both sides by the coefficient if x2. This gives x2 -10x +24.5 = 0.
Second step is to forget about the constant 24.5 for now. Let's look at the expression x2-10x+C and try to find a value of C that makes this a perfect square (x+D)2. Next, note that since the coefficient of x is negative, D must be negative. Also note that the coefficient of x, -10, must be twice D. This means the expression X+D must be equal to x-5.
So now we have (x-5)2 = x2-10x+25 which is a perfect square, and if we multiply it by 2, we get 2x2-20x+50 which matches the first two terms of the original equation. How do we use this to solve the original equation?
Let's re-write the original equation now. we started with 2x2 - 20x + 49 = 0. Let's re-write this using our completed square as 2x2 - 20x + 50 - 1 = 0. Now, since the first three terms form a perfect square multiplied by 2, we have 2(x-5)2 -1 = 0.
Then 2(x-5)2 = 1, so (x-5)2 = 1/2, and (x-5) = +1/√2 or -1/√2, and x = 5+1/√2 or 5-1/√2
Rule: if you are using the method of completing the square to find the zeros of a quadratic expression, it means you should
1. forget about the constant and force the remainder of the expression to be a square, then
2 substitute the completed square back into the original expression - the original equation is the completed square plus or minus a constant remainder, and finally,
3 move the constant remainder to the other side of the equation and take the square root of both sides. Remember that a square root can be positive or negative which creates two cases.
5. solve each of the two cases for x.
