Ch 12 notes

Algebra I Chapter 12 Notes Mrs. Cataldo Chapter 12 Notes 12.1 Squares and square roots In the equation y = x², y is the __________ and x is the ________________________. The symbol is called a ______________. NOTE: The square root of a number can be either negative or positive. (What are the square roots of 36?) ______ By convention, the symbol means the POSITIVE sq. rt. and we write the NEGATIVE sq. rt. as ‐. When a number such as 50 has a square root that is not an integer, we have to approximate it. What is the approximate square root. of 50 to the nearest tenth? ____ the nearest hundredth? _______ Look at page 563. Mark it and use it to answer problems in this chapter. You can use your calculator for problems with an asterix. Can a negative number have a square root? __________ Examples: Set 2 #5 12.2 Square Roots of Products According to the law of exponents, what does _________ If that is true, then € 2 x 12       = x1 and is another way of writing . When x is not negative. We can see that € 1(xy)2= so we can see that = . We use this rule to “simplify” square roots. If we can factor the number or expression under the radical sign into products containing squares, we can remove them from under the radical sign. Example: “Simplify” When a radical is simplified it is said to be in __________________________. That is, the integer has no factors that are squares of integers other than 1 and ‐1. This is also true for a monomial (assuming the variables represent nonnegative numbers). The square root of a monomial is in ___________________________ if the monomial has no factors that are squares of monomials other than 1 or ‐1. Example: “Simplify” When doing homework you will be referring back to the square root table on page 563. Examples: Set 2 4f,g,m, 5f, 6g,h, 7b, 10e 12.3 Square Roots of Quotients Since we know that , we know that . Using this rule we can simplify fractions to remove radicals from the denominator. If there are no radicals in the denominator and the numerator is in simple radical form then the whole expression is in simple radical form. Example: Simplify . If we can make the denominator a square by multiplying it by another integer, we can take it’s square root and remove the radical sign. We know that and that 3 x 12 = 36 which is a square so we can write which is in simple radical form. Examples: p. 574 Set 2 6e,f, 7d, 9f Short Quiz: Simplify 1. 2. 3. 4. 5. 12.4 Adding and Subtracting Square Roots Look at the spiral on page 577. Square roots can be used to mathematically describe patterns in nature. See that in general Although you can’t add numbers under a radical, you can simply square roots if you factor and simplify each term first. eg. Simplify You can’t add to get but you can simplify and Now can add € 2 2 + 3 2 = Because the number under the radical for each term is the same. Examples: p. 579 Set 2 4c, 5d,e, 8g, 9a 12.5 Multiplying Square Roots We cannot add square roots as follows, but we can multiply them since then, eg. Simplify: 4 (4· 5)·( ) = 20 = 20x Examples: Multiply and simplify 1) (3‐ ) 2) € (5 + 2)(1+ 3 2) 3) € (3 x3 )(9 x9 ) 4) € (6 + 5)(2 + 2 5) 12.6 Dividing Square Roots When we have a fraction with a square root as part of the denominator, we need to rationalize the denominator, i.e. remove the square root from the denominator. A conjugate of a binomial is the binomial that you can multiply it by to eliminate any square roots. What are the conjugates of the following binomials? _____________ _____________ _______________ In the example from the your book, the Parthenon is based on a golden rectangle such that the length is times the width. Rationalize the denominator and estimate the multiplication factor. To get rid of the square root we need to multiply the denominator by the binomial that will square the square root and eliminate any middle terms, also called the _____________________. What might that be? ______________ The simplified version of this fraction is __________________________________________ Example: Rationalize the denominator. Solve for x. 12.7 Radical Equations In this section we will use formulas again. This time they will contain radicals. They are called _______________ equations. The speed a cyclist can turn a corner is given as where s is speed in mph and r is the radius of the corner in feet. If a cyclist is going 30 mph, what is the radius of the sharpest corner he can take without wiping out? s = 30 miles/hour so 30 = 4 solve for r by dividing both sides by ____ and then squaring both sides. € 30 4 = ( € 30 4 )² = r r = (7.5)² = 56.25 feet Note that when solving for a variable, you can square or take the square root of both sides of the equation and have an equivalent equation. Examples: Solve the following radical equations 1) € 2 x + 4 = 0 2) € x + 8 = 3 2 3) € 6 − x =11