Example Card：Solving $ax^2 + bx = c$ by 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
What if the first term has a coefficient? There's just one extra step to clear it out. This example covers the key idea of handling quadratic equations where the $x^2$ term isn't 1.
Solve the equation by completing the square: $2x^2 + 12x = 14$.
1. The coefficient of the quadratic term must be $1$. Divide both sides of the equation by $2$. $$\frac{2x^2 + 12x}{2} = \frac{14}{2}$$ 2. Simplify the equation. $$x^2 + 6x = 7$$ 3. Now complete the square. Add the square of half the coefficient of the $x$ term to both sides. $$x^2 + 6x + \left(\frac{6}{2}\right)^2 = 7 + \left(\frac{6}{2}\right)^2$$ 4. Simplify the term inside the parentheses and then simplify the full expression. $$x^2 + 6x + (3)^2 = 7 + 9$$ 5. Factor the left side and simplify the right side. $$(x+3)^2 = 16$$ 6. Take the square root of both sides. $$\sqrt{(x+3)^2} = \pm\sqrt{16}$$ 7. Simplify the equation. $$x+3 = \pm 4$$ 8. Write the two possible equations and solve for $x$. $$x+3 = 4 \quad \text{or} \quad x+3 = 4$$ $$x = 7 \quad \text{or} \quad x = 1$$.
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.