# 3.5 – Completing the Square

## Objectives

• Complete the square algebraically to find the solutions to a quadratic equation.
• Complete the square with tiles.
• Determine the number that must be added to an expression to complete the square.

## Key Terms

• Complete the Square – A way to solve quadratic equations. It involves adding a number to both sides of an equation to make one side a perfect square trinomial.
• The equation can then be solved by taking the square root of both sides and simplifying.
• Perfect Square Trinomial – The result of multiplying a binomial by itself (squaring it).
• Ex. $x^2-10x+100=(x-10)(x-10)=(x-10)^2$

## Notes

Solving Quadratics with Perfect Square Trinomials
• How to Solve a Quadratic Equation with a Perfect Square Trinomial
• Step 1: Factor the perfect square trinomial.
• Step 2: Take the square root of each side of the equation.
• Step 3: Simplify the resulting equations.
• Step 4: Solve for x in each equation.

Completing the Square
• Formula:  Add $(\frac{b}{2})^2$ to both sides

• Remember: “b” is the middle term in standard form (the x-term)

• Remember: what you do to one side, you MUST do to the other (to balance the equation)
• Algebra Tiles – A visual model of a “Perfect Square” trinomial

• Example: $x^2=100$ has 2 solutions (one positive and one negative)
• $(\sqrt{x})^2=x$ and $\sqrt{100}=\pm10$
• So, x = 10 or x = -10