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Unit 10 Radicals and Quadratic Equations • Learn the Three “Nevers” for Radicals • Simplify Radical Expressions • Solve Quadratic Equations Lecture 1 Objectives • Define a radical expression • Simplify perfect square factors under the radical • Multiply radicals Properties of Radicals Radical Index y x Radicand What number multiplied by itself “y” times gives “x”? Example Simplify: 2 25 5 49 7 8 42 4 2 2 2 Simplifying Radical Expressions 1. No perfect squares under the radical 2 24 46 4 6 2 6 Factor out perfect squares! Multiplying Radicals • Radicals with like indices can be multiplied. 3 5 3 7 35 3 3 18 54 9 6 3 6 Homework Set 10.1 • WS Simplifying Radicals 10.1 Lecture 2 Objectives • Simplify perfect Do you know square factors under what I am? the radical • Multiply radicals • Work with fractions I am a Perfect Square! and radicals Simplifying Radical Expressions 2. No fractions under the radical Convert a 2 2 fraction under the radical into a fraction of 3 3 radicals! Simplifying Radical Expressions 3. No radicals in the denominator 2 3 6 6 3 3 9 3 Multiply numerator and denominator by the radical in the denominator! Homework Set 10.2 • WS Simplifying Radicals 10.2 • Quiz next class day Lecture 3 Objectives • Define a quadratic equation • Solve quadratic equations by taking square roots Quadratic Equation A quadratic equation is like a linear equation except that it contains a variable raised to the second power. Quadratic term Linear term ax bx c 0 2 Constant term Solving a Quadratic Equation Solving a quadratic equation is much more complicated than solving a linear equation. Try to solve the following equation using the properties of equality: x 3x 4 0 2 Solving a Quadratic Equation While the properties of equality cannot solve most quadratic equations, they do work on some of them: 2x 8 0 2 Solving a Quadratic Equation by Square Roots Any quadratic equation the lacks a linear term can be solved by: 1. Solving for the squared variable using the properties of equality 2. Taking the square root of both sides of the equation to solve for the variable 3. Make sure to include the “plus or minus” with the square root step, because the square root of a number has two answers. ax bx c 0 2 Example Solve: 3x 24 0 2 3x 24 2 x 8 2 x 8 2 x 2 2 A few more complicated quadratics seem to be all “set-up” for square roots: ( x 7) 162 ( x 7) 16 2 x 7 4 x 7 4 3 or 11 Homework Set 10.3 • WS Quadratic Equations Square Roots 10.3 Lecture 4 Objectives • Solve quadratic equations by factoring Solving a Quadratic Equation Many quadratic equations can be solved by factoring them into a product, and then using something called the zero-product rule. x 3x 0 2 Factor out the “x” x x 3 0 Apply the zero-product rule x 0 or 3 Viola! The Zero-Product Rule The only way a set of numbers can multiply to zero is if one of the numbers is zero. If a b 0, then a 0 or b 0. While not all quadratics can factor into products, the zero-product rule is very useful to solve those that do. Examples ( x 7) 0 Solve: 2 x 7 (4 y 3)( y 2) 0 y 3 or 2 4 z (3z 2)( z 2) 0 z 0 or 2 or 2 3 Factoring Quadratics Today we will focus on two kinds of factoring: 1. Factoring out a monomial. 2. Factoring a trinomial square. ax bx c 0 2 Factoring Out a Monomial When there is no constant term in a quadratic, you can always factor out a common monomial factor: 3x 4 x 0 2 x(3 x 4) 0 Then use the zero-product rule to get the solutions. Trinomial Squares Some trinomial quadratics (has all three terms) factor into perfect squares. The only way they do this, however, is if their coefficients have a special relationship given by these forms: a 2ab b a b 2 2 2 a 2ab b a b 2 2 2 Trinomial Squares Notice that the first and last terms are perfect squares, and the linear term is two times the product of the square roots of the ends. Only very special quadratic equations can be factored this way. a 2ab b a b 2 2 2 a 2ab b a b 2 2 2 Examples Which of these are trinomial squares? x 4x 4 0 2 yes 9 x 24 x 16 0 2 yes 4x 6x 9 0 2 no 25x 70 x 49 0 2 yes Solving Trinomial Squares Trinomial squares are the easiest of all the quadratics to solve. Simply factor them according to their form, and then use the zero- product rule. a 2ab b a b 2 2 2 a 2ab b a b 2 2 2 Examples Solve: 9 x 24 x 16 0 2 (3x 4) 0 2 x4 3 4 x 12 x 9 0 2 (2 x 3) 0 2 x 3 2 Homework Set 10.4 • WS Quadratics 10.4 Lecture 5 Objectives • Solve quadratic equations by factoring Factoring Quadratics Today we will focus on another kind of factoring. Many quadratic polynomials are the result of multiplying two binomials together: x 3 x 2 x 2 5x 6 x 7 x 3 x 2 4 x 21 x 1 x 5 x 2 4 x 5 Factoring Quadratics We can reverse this process and factor many quadratics where the quadratic coefficient is = 1 by doing the following: x bx c x m x n 2 where b m n and c m n All you have to do is find two numbers that multiply to “c” and add to “b”! Examples Factor: x 3x 2 x 1 x 2 2 y y 20 2 y 5 y 4 r 5r 6 r 3 r 2 2 z 3z 70 z 10 z 7 2 Factoring Quadratics After these quadratics are factored, use the zero-product rule to find the solutions: x 5 x 24 x 8 x 3 2 x 8 or 3 Homework Set 10.5 • WS Quadratics 10.5 Lecture 6 Objectives • Solve quadratic equations by factoring Factoring Quadratics Today we will focus on another kind of factoring. Many quadratic polynomials are the result of multiplying two binomials together: 2 x 3 x 2 2 x 2 7 x 6 3x 7 2x 3 6 x2 5x 21 5x 13x 5 15x2 22 x 5 Factoring Quadratics We can reverse this process and factor many quadratics where the quadratic coefficient is 1 by doing the following: 1. Multiply a x c and write it above b. 2. Find two numbers “m” and “n” so that m x n = a x c and m + n = b. 3. Rewrite the quadratic but replace the linear term with two terms with coefficients “m” and “n”. 4. Factor by grouping. ax bx c 0 2 Factor: 6 Example Rewrite the original 2x 7 x 3 2 Multiply a x c and write above b equation replacing b m 6 and n 1 Find m x n = a x c and m + n = b 2 x 6 x 1x 3 2 Group and factor 2x 6x 1x 3 2 2x x 3 1 x 3 Use the Distributive 2x 1 x 3 Property Homework Set 10.6 • WS Quadratics 10.6 • Quiz next class day Lecture 7 Objectives • Review Quadratics • Learn the Factoring Map Solving Quadratics Quadratic equations contain a variable squared and cannot be solved by using the properties of equality. To solve a quadratic, several different methods need to be used: 1. Take square roots 2. Factor: 1. Simple monomial 2. Trinomial square 3. a=1 (short-cut) 4. a1 (4-step) Homework Set 10.7 • WS Quadratics 10.7

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posted: | 11/23/2011 |

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