Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form of a perfect square trinomial.

The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$ , where $$a$$ , $$b$$ , and $$c$$ are constants.

1. Move $$c$$ to the other side: $$x^2 + bx = c$$ . 2. Take half of $$b$$ , square it, and add to both sides: $$x^2 + bx + (b/2)^2 = c + (b/2)^2$$ . 3. Factor the left side.

The vertex form of a quadratic equation is $$y = a(x - h)^2 + k$$ , where (h, k) is the vertex of the parabola.

$$x = 1, x = -15$$ .

$$x = 19, x = -3$$ .

Complex solutions occur when the discriminant (b^2 - 4ac) is negative, indicating that the parabola does not intersect the x-axis.

$$x = 3 ± i√82$$ .

$$x = -1 ± i√19$$ .

The discriminant is given by $$D = b^2 - 4ac$$ and determines the nature of the roots of the quadratic equation.