Solving by Completing the Square
Solve quadratic equations by completing the square: add the square of half the linear coefficient to both sides, factor the perfect square trinomial, then apply the square root property.
Key Concepts
This method transforms a quadratic equation into a form that can be solved using the Square Root Property. The process involves isolating the variable terms, adding a constant to both sides to form a perfect square trinomial, factoring it, and then taking the square root of both sides to find the solutions for the variable.
Solve $x^2 8x 9 = 0$. First, $x^2 8x = 9$. Add $(\frac{ 8}{2})^2=16$ to both sides: $x^2 8x + 16 = 9 + 16$. This gives $(x 4)^2 = 25$, so $x 4 = \pm 5$, and $x=9$ or $x= 1$. Solve $x^2 + 6x 2 = 0$. First, $x^2 + 6x = 2$. Add $(\frac{6}{2})^2=9$ to both sides: $x^2 + 6x + 9 = 2 + 9$. This gives $(x+3)^2 = 11$, so $x = 3 \pm \sqrt{11}$.
This is your go to strategy when a quadratic equation refuses to be factored easily. First, you shove the constant term to the other side. Then, you work your magic by 'completing the square' on the variable side, making sure to add the same value to the other side to keep the equation balanced. Finally, you use the Square Root Property to win the game.
Common Questions
What are the steps for completing the square to solve a quadratic?
Move the constant to the right side. Take half of the linear coefficient b, square it, and add to both sides. The left side is now a perfect square trinomial; factor it as (x + b/2)^2. Take the square root of both sides with plus or minus, then solve for x.
When should you use completing the square instead of the quadratic formula?
Completing the square is useful when the equation is near perfect square form, when deriving vertex form of a parabola y=a(x-h)^2+k, or when the quadratic formula is not yet memorized. It is also the method used to derive the quadratic formula itself.
How does completing the square connect to vertex form of a parabola?
Completing the square on ax^2+bx+c rewrites it as a(x-h)^2+k where (h,k) is the vertex. This reveals the maximum or minimum point directly without graphing, making it essential for optimization problems in Grade 10 Saxon Algebra 2.