Example Card：Solving $x^2 + bx = c$ 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
Let's transform this equation into a perfect square to make solving it a breeze. This first example shows the core technique of completing the square when the quadratic term is already simplified.
Solve the equation by completing the square: $x^2 + 12x = 13$.
1. First, we need to complete the square on the left side. We add the square of half the coefficient of the $x$ term to both sides. $$x^2 + 12x + \left(\frac{12}{2}\right)^2 = 13 + \left(\frac{12}{2}\right)^2$$ 2. Simplify the term in the parentheses. $$x^2 + 12x + (6)^2 = 13 + (6)^2$$ 3. Simplify the expression. $$x^2 + 12x + 36 = 13 + 36$$ 4. Factor the perfect square trinomial on the left and simplify the right side. $$(x+6)^2 = 49$$ 5. Now, take the square root of both sides to solve for $x$. $$\sqrt{(x+6)^2} = \pm\sqrt{49}$$ 6. Simplify the equation. $$x+6 = \pm 7$$ 7. Write the two possible equations and solve each one. $$x+6 = 7 \quad \text{or} \quad x+6 = 7$$ $$x = 13 \quad \text{or} \quad x = 1$$.
Common Questions
How do you solve an equation of the form x² = a?
Take the square root of both sides and remember to include both the positive and negative roots. The solution is x = ±√a. Simplify the radical if possible, or leave it in exact form.
Why do equations of the form x² = a have two solutions?
Squaring a positive or negative number gives the same positive result, so both x = √a and x = -√a satisfy x² = a. Missing the negative root is the most common error when solving this type of equation.
What happens when a is negative in x² = a?
If a is negative, there is no real solution because no real number squared produces a negative result. You would write 'no real solution.' This connects to even roots of negative numbers being undefined in the real number system.