Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting them in the form of a perfect square trinomial, allowing for easier solutions.

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

Move the constant term to the other side: $$x^2 + 6x = 5$$ becomes $$x^2 + 6x - 5 = 0$$ .

Take half of the coefficient of $$x$$ (which is 6), square it (resulting in 9), and add it to both sides of the equation.

The completed square form is $$(x + 3)^2 - 14 = 0$$ .

The new C-term is calculated as $$C = \frac{b}{2}^2$$ .

The solutions are $$x = 10$$ and $$x = -4$$ .

Rearrange to $$x^2 + 12x - 5 = 0$$ , complete the square to get $$(x + 6)^2 - 41 = 0$$ , then solve for $$x$$ .

The vertex represents the maximum or minimum point of the parabola, depending on the direction it opens.

If the discriminant (b^2 - 4ac) is positive, the quadratic equation has two distinct real solutions.