Completing the Square & Quadratic Formula

The quadratic formula is used to find the solutions of a quadratic equation in the form $$ax^2 + bx + c = 0$$ . It is given by $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ .

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form $$(x - p)^2 = q$$ , which makes it easier to solve for x.

The discriminant is the part of the quadratic formula under the square root, given by $$b^2 - 4ac$$ . It determines the nature of the roots of the equation.

A positive discriminant indicates that the quadratic equation has two distinct real roots.

A zero discriminant indicates that the quadratic equation has exactly one real root, also known as a repeated or double root.

A negative discriminant indicates that the quadratic equation has two complex (imaginary) roots.

1. Move the constant to the other side: $$x^2 + 6x = -5$$ . 2. Take half of the coefficient of x (which is 3), square it (9), and add to both sides: $$x^2 + 6x + 9 = 4$$ . 3. Factor: $$(x + 3)^2 = 4$$ . 4. Solve: $$x + 3 = \pm 2$$ , giving $$x = -1, -5$$ .

Identify the coefficients a, b, and c from the quadratic equation in the form $$ax^2 + bx + c = 0$$ .

Using the quadratic formula: a = 2, b = 4, c = -6. The solutions are $$x = \frac{{-4 \pm \sqrt{{4^2 - 4(2)(-6)}}}}{{2(2)}} = \frac{{-4 \pm \sqrt{{16 + 48}}}}{{4}} = \frac{{-4 \pm 8}}{{4}}$$ , giving $$x = 1, -3$$ .

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