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Section 8.1 (Solving Quadratic Equations by Completing the Square) In chapter 5, we solved quadratic (2nd degree) equations by solving for 0 and factoring (x2 = 9) Not all quadratic equations can be solved using this method (m2 – 7m – 1 = 0 cannot be factored and solved) so we need to explore other methods Consider solving the previous equation by taking the square root of both sides (x 2 = 9 again) Square Root Property => If b is a real number, and if a2 = b, then a = b Examples: Use the square root property to solve x2 = 20 5x2 – 55 = 0 (x + 2)2 = 18 (3x – 1)2 = -4 Notice in the last 2 examples that if we have the square of a binomial on one side, we can solve the equation (a squared binomial is the factored form of a perfect square trinomial) If we can achieve a perfect square trinomial on one side of the equation, we can solve it In perfect square trinomials, the constant (last) term is the square of half of the middle term o (x+3)2 = x2+6x+9, where ½(6)=3 and 32=9 (x-4)2 = x2-8x+16, where ½(-8)=-4 and (-4)2=16 The process (getting the perfect square trinomial on 1 side to solve) is called completing the square 1. If the coefficient of the x2 (first) term is 1, go to the next step; otherwise, divide both sides by that coefficient 2. Isolate all variable terms (terms with letters) on one side of the equation 3. Complete the square by adding ½ the coefficient of the x (middle) term to both sides 4. Factor the resulting perfect square trinomial and write as a square 5. Use the square root property to solve Examples: Solve by completing the square x2 + 8x = 1 y2 – 5y + 2 = 0 5x2 - 10x + 2 = 0 2x2 – 2x + 7 = 0 The formula for calculating the amount of money when interest is compounded annually is A = P(1 + r)t where P is the original investment (principal), r is the interest rate per compounding period (per year for annual), and t is the number of periods (number of years for annual). Example: Use this formula to find the interest rate r if $1600 compounded annually (at Sesame Street National Bank) grows to $1764 in 2 years. (see book pp. 484-486 for good discussion on simple/compound interest and how it is calculated)