Completing the square

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial, making it easier to solve or analyze.

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

Take half of the coefficient of x (which is -4), square it, resulting in 4. Thus, $$x^2 - 4x + 4 = (x - 2)^2$$ .

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

The vertex can be found using the formula $$h = -\frac{b}{2a}$$ and $$k = f(h)$$ , where f(h) is the value of the quadratic at x = h.

The discriminant, given by $$D = b^2 - 4ac$$ , 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.

The ± symbol indicates that there are two possible solutions when taking the square root of both sides of an equation.

First, divide by 2: $$x^2 - 10x + 9 = 0$$ . Then, complete the square: $$x^2 - 10x + 25 = 16$$ , leading to $$(x - 5)^2 = 16$$ .

Rearrange the equation to standard form: $$3x^2 - 18x + 15 = 0$$ , then divide by 3 to simplify.

Divide the b value by 2 and square the result to find the constant needed to complete the square.