Lesson 10

# Lesson 10 - Lesson 10 Quadratic equation an equation of the...

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Unformatted text preview: Lesson 10 Quadratic equation: - an equation of the form are real numbers and - special forms of quadratic equations: , where and The degree of a quadratic equation is ________ Example 1: Determine whether each of the following equations is quadratic. a. b. c. d. ) e. ( f. g. In order for an equation to be quadratic, the degree must be _________. Example 2: If , what must be true about and ? Example 3: If , what must be true about , and ? Zero Factor Theorem: - the product of two or more factors is zero if and only if at least one of the factors is zero if and only if or if and only if , or Does this work for any values other than zero ( or Why not? )? Why Solving an equation by factoring: 1. write the equation in polynomial form and set it equal to zero 2. factor 3. use Zero Factor Theorem Example 4: Solve the following equations by factoring: a. ) b. ( c. d. e. Is there another way to solve parts d. and e.? If get just , what could we do to both sides of the equation to by itself? √ √ Special Quadratic Equation: - to solve an equation that involves a perfect square, take the square root of both sides - this can be used for a single term that is a perfect square, or a quantity that is a perfect square - if , then √ ) - if ( , then √ and √ Do we have to include the sign? Why or why not? Example 3: Solve the following equations by taking the square root of both sides a. b. ( c. ( ) ) d. ( ) Why is it that if ? √ , then , but if , then We’ve seen how to solve quadratic equations which are factorable and quadratic equations which are perfect squares, but what about quadratic equations which are neither? If a quadratic equation is not factorable and is not a perfect square, how do we solve it? Quadratic equations which are not factorable and are not perfect squares can be changed into perfect squares by using a procedure called completing the square. Examples of Perfect Squares: ( ) ( )( ( ) ( ) ( ) ( ) ( ) ) When a perfect square is in quadratic form, do you notice a relationship between the coefficient of the middle term and the constant term? The constant term is always half the coefficient of the middle term squared. ( ) ( ) ( ) ( ( ) () ( ) ) () ( ) ) ( ( ( ) ) Using the pattern from the previous examples, how can we make a quadratic equation such as a perfect square? Completing the square: Given a quadratic equation of the form that is not factorable, 1. divide each term by the leading coefficient ( ) 2. isolate the constant term ( ) 3. add the square of half the coefficient of to both sides () 4. solve using the special case ( If the leading coefficient is 1 ( √) ), skip the first step. Example 4: Solve the following equations by completing the square: a. b. c. ...
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