Recall the expansion for a square binomial, and think about how can you get an arbitrary expression to that form.
See solution.
Practice makes perfect
We can solve an equation of the form x^()2 + bx = c, where b and c are real numbers, by completing the square. Notice that we can always rewrite any quadratic equation in this form by isolating the variable terms and dividing by the coefficient of x^()2 if needed. Let's see an example.
Now we need to ensure that we have an expression that is a perfect square trinomial. Then, we will be able to factor it as the square of a binomial and take the square root on both sides to find the solutions. Let's recall the expansion of the square of a binomial.
(x+a)^2 = x^2+2ax +a^2By adding the appropriate constant, we can obtain a perfect square trinomial. Let's compare the right-hand side of this formula to the left-hand side of our expression.
x^2+2ax +a^2
x^2+bx + ?
Notice that the b parameter of our expression corresponds to the factor 2a in the expansion for the square of the binomial. We can find the quantity corresponding to a^2 that we would have to add from this relationship by solving for a first and then squaring both sides.
This means that by adding the term ( b2)^2 we can get a perfect square trinomial! Let's try this with the example equation we had at the beginning, x^2+6x =- 8. Notice that here b=6.
