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 ≠ 0.

1. Move 7 to the left: $$x^2 - 6x - 7 = 0$$ . 2. Take half of -6, square it: $$(-3)^2 = 9$$ . 3. Add 9 to both sides: $$x^2 - 6x + 9 = 16$$ . 4. Factor: $$(x - 3)^2 = 16$$ .

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

The first step is to move the constant term to the other side of the equation.

Take half of 8 (which is 4) and square it (4^2 = 16). Add 16 to both sides of the equation.

1. Move -5: $$x^2 - 4x = 5$$ . 2. Complete the square: $$x^2 - 4x + 4 = 9$$ . 3. Factor: $$(x - 2)^2 = 9$$ . 4. Solve: $$x - 2 = \pm 3$$ , so $$x = 5, -1$$ .

A quadratic has complex solutions when the discriminant (b^2 - 4ac) is less than zero, indicating that the parabola does not intersect the x-axis.

Starting from the standard form $$ax^2 + bx + c = 0$$ , divide by a, move c/a, complete the square, and solve for x to derive $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ .

The discriminant is the expression $$b^2 - 4ac$$ that determines the nature of the roots of the quadratic equation.