Completing the Square Practice

To complete the square for a quadratic equation of the form $$ax^2 + bx + c$$ , rewrite it as $$a(x + \frac{b}{2a})^2 - \frac{b^2 - 4ac}{4a}$$ .

The value to be added is $$9$$ , calculated as $$\frac{6}{2}^2$$ .

Take the square root of both sides to get $$x + 2 = \pm 10$$ .

The completed square form is $$(x + 4)^2 = 25$$ .

The next line is $$(x - 2)^2 = 9$$ .

You should add $$36$$ , calculated as $$\frac{-12}{2}^2$$ .

Completing the square allows us to rewrite the quadratic in vertex form, making it easier to analyze its properties such as vertex and axis of symmetry.

The vertex is the point $$(h, k)$$ .

The coefficient $$a$$ determines the direction of the parabola (upward if $$a > 0$$ , downward if $$a < 0$$ ), while $$b$$ and $$c$$ affect the position and shape.

The discriminant $$D = b^2 - 4ac$$ indicates the nature of the roots: if $$D > 0$$ , there are two distinct real roots; if $$D = 0$$ , there is one real root; if $$D < 0$$ , there are no real roots.