Download Section 8.1

Section 8.1 (Solving Quadratic Equations by Completing the Square)
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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)
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Square Root Property => If b is a real number, and if a2 = b, then a =
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 b
Examples: Use the square root property to solve
x2 = 20
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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
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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)