Completing the Square
Complete the square to convert quadratic expressions and solve equations in Grade 9 Algebra. Add (b/2)² strategically to create a perfect square trinomial.
Key Concepts
Property To complete the square for an expression like $x^2 + bx$, you add a special value: $(\frac{b}{2})^2$. This action transforms the expression into a perfect square trinomial, which factors neatly into $(x + \frac{b}{2})^2$ and makes it easier to handle. Explanation This method is like a puzzle! You are finding the one missing piece that turns your expression into a perfect, balanced square. This makes it way easier to work with when you are solving the full quadratic equation later on. It’s a foundational step for solving quadratics! Examples $x^2 + 12x \rightarrow x^2 + 12x + (\frac{12}{2})^2 = x^2 + 12x + 36 = (x+6)^2$ $y^2 10y \rightarrow y^2 10y + (\frac{ 10}{2})^2 = y^2 10y + 25 = (y 5)^2$.
Common Questions
What is the process of completing the square?
Move the constant to the right side, then add (b/2)² to both sides to make the left side a perfect square trinomial. Factor the left side as (x + b/2)², then take square roots of both sides and solve for x.
Why do you add (b/2)² when completing the square?
Adding (b/2)² creates the missing term that turns x² + bx into a perfect square trinomial (x + b/2)². This value comes from the pattern (x + d)² = x² + 2dx + d², where d = b/2.
When is completing the square more useful than the quadratic formula?
Completing the square is especially useful for converting a quadratic to vertex form y = a(x-h)² + k, which reveals the vertex directly. It is also the derivation behind the quadratic formula itself.