MAT029C. 8.2 Multiplying and Simplifying with Radical Expressions.

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1 MAT029C TOPICS: RADICAL EQUATIONS QUADRATIC EQUATIONS Radical Equations 8.1 Introduction to Radical Expressions I. Basically, squaring and square rooting undo each other: = 9 and = 3. So, = 9, right? II. If radicals seem slippery to you, you re right! Note these fine points: A. The words square roots usually yield 2 numbers but the symbol, only one. Square roots of 4 = 2, -2 but. The symbol yields the positive one. B. Consider. A simplification is exact. An approximation ( is not exact. C. is a real number (4) but is not real. (How could any squared number be negative?) D. Consider and. They both are 3. That means absolute value. So, if you are squaring and square rooting any number, you must mention this. BUT if the instructions tell you that the a-stuff is not negative, you can forget the absolute values. Later you ll have to be careful about this. E. If you can simplify a radical. You must! III. The radicand is the stuff under the. IV. Memorize the first 12 perfect squares today so you can hunt for them. 8.2 Multiplying and Simplifying with Radical Expressions. I. Pull apart Rebundle If instructions say multiply you may simplify and then multiply OR multiply and then simplify. II. Simplifying Radicals Pulling apart to start the process A. A number 1. Look for largest perfect square factor (4,9,16,25,36,49,64, 81,100 ) = 4 2. Use a factor tree to get primes and then recall that pairs are squares. B. A variable 1. If power is even, cut in half. (Recall pairs are squares.) 2. If power is odd, pull off one and then cut in half 1

2 C. A Bi-Tri-nomial Factor to see if you can get stuff to powers III. Simplifying Radicals Rebundling to Finish 8.3 Quotients Involving Radical Expressions I. Pull apart Rebundle If instructions say Divide You may simplify and then divide OR divide and then simplify. II. Simplifying is very similar to 8.2 but watch the bottom III. A final answer may not have a Downstairs!!! If you have one you can fix it by multiplying by a fancy one composed of denominator radical over itself. (Note how you see the troublemaker three times when you do this.) 8.4 Addition, Subtraction, and More Multiplication, (etc.) I. To add or subtract radicals, they must be like. If they aren t, try simplifying until they are: II. Multiplying A. Simple B. Distribution Style C. Foil Style III. Dividing A. Dividing here just means Rationalizing the Denominator 1. Simple 2. Fancy: If you see a binomial form downstairs, you need its conjugate Identical except sign in middle 8.5 Radical Equations Cancel I. The intuitive way to get rid of a square root in an equation if to square it. Make sure you check. (Just as in clearing, the denominator, you must check.) II. One Radical Isolate, Square, Check 0 Pull apart Pull Rebundle apart Rebundle X 5 = 0 so x = 5 X 1 = 0 so x = 1 2 If instructions

3 III. Two Radicals You must split them up and you will have to square twice. Check: Good one! 8.6 Applications with Right Triangles I. The minute you hear mention of a right triangle, write the Pythagorean Theorem: with c= the hypotenuse or longest side. II. Either the hypotenuse or a leg is unknown. A. a=3 c b=5 B. c=13 a b=12 3

4 III. This is one of the very few places that Usually Here we disregard the negative answer because we are dealing with lengths. Watch out for bad habits!!!! 9.1 Introduction to Quadratic Equations I. Quadratic equations have the unknown squared either at the beginning or after some multiplication. They are not solved as linear equations. There are several methods II. Most methods use Standard Form (.) and then setting each factor to zero. To get standard form, you may need to multiply (clearing, dist, FOIL ) and move terms. 5(t - 3) 30 = (t 3)(t + 3) (t-3)(t+3) ( )( ) 5t = 9 - = 1 (t - 3)(t + 3) 5t 15 = 9-5t t + 15 (t-3)(t+3) - 5t + 6 = 0 III. Factor and find: Method 1 of 4 (If it doesn t factor, try something else ) (t 3)(t 2) = 0 t 3 = 0 t 2 = 0 t = 3 t = Solving quadratic equations by completing the square I. Before Completing the Square, if have and no x or the only x is in parens and squared, just do this Method 2 of x = -6, 4?? II. Completing the square is a method that always works, but it s easy to forget = 1 Step 1: Divide all by a if not 1 Step 2: Move constant to right of = leaving a spot where it was. Step 3: Take and add to both sides Step 4: Factor left; simplify right using root of above Step 5: Step 6: Finish Method 3 of 4 = 4

5 9.3 The Quadratic Formula I. This Method 4 of 4 always works, but you must memorize it and simplify its results carefully. x = II. Simplifying Results A. Watch the radical (A funny way to memorize this is using the song, Pop Goes the Weasel.) is not done! is. B. Watch for factoring. is not done! = is. III. If a negative pops up under the radical, you have no real # solution. 9.4 Formulas I. Formulas often have lots of variables and you will be told to solve for a particular one. If you are given one that has a downstairs (and no ), clear it. S = 3bS = a + b II. How you proceed depends on whether your unknown has an invisible power of one (linear), a power of two (quadratic), or a radical). A. Linear: Dogs/Cats/Mice/ Frickin /Insects to remember Dist/Clean/Move/Factor/Isolate. B. Quadratic: Four methods above. C. Radical: Square both sides and check. +/- 9.5 Applications and Problem Solving I. Please Let Frankie Sing! Picture, Label, Formulas, (Translate), Substitute. A. Area formulas (back cover) B. Pythag ( C. Uniform edge around rectangle. Label with x all around Inside rectangle area is (12 2x)(15 2x) II. Rate x Time = Distance This is a twist on 7.5 because you get total time. A. Trick: If you have total time as 3 hours, you can call one time t and the other 3-t because they total 3. Get it? B. Honest way: Call one time t 1, and the other, solve for each, add them and set to total 5

6 9.6 Graphs of quadratic equations I. Equations that have two unknowns must be graphed. They have some common features with straight lines: A. Table of values that you select and calculate: x y B. Intercepts: (0, )(,0) give nice anchors to the graph. II. Quadratic equations have some unique features A. Shape is parabola 1. If a is positive, shape is like this: 2. If a is negative, shape is like this:, B. The vertex is the point where the parabola changes direction. It is located at. [Once you have the x, plug it in to get y.] 6