ALG 1B/ cdipaulo2 In this chapter we will look at… 9.1 Square Roots 9.2 Solving Quadratic Equations –By finding square roots 9.3 Simplifying Radicals 9.4 Graphing Quadratic Functions 9.5 Solving Quadratics by Graphing 9.6 Solving Quadratics using the Quadratic Formula 9.7 The Discriminant 9.8 Graphing Quadratic Inequalities
ALG 1B/ cdipaulo3 Watch this… Write 5 important ideas from the clip. We will be sharing these in class.
ALG 1B/ cdipaulo4 Square Roots 9.1 How many squares are on each side of a chess board?
ALG 1B/ cdipaulo5 Square Roots All positive real numbers have 2 square roots. one positive square root one negative square root Square Roots are written using a radical sign. The Number inside the radical sign is the radicand. Square Roots Undo Numbers Squared. 3 2 = 33= 9 therefore V9 = 3 the positive square root. 9 also = -3 the negative square root, because (-3)(-3) = 9 ± 9 means the positive and negative square roots of 9.
ALG 1B/ cdipaulo6 Perfect Squares Recall Irrational Numbers include the square root of any non perfect square. Perfect Square are rational numbers. –They are integers. The square of an integer is a perfect square. n ±n±1±2±3±4±5±6±7±8±9±10±11± =12 2 =43 2 =94 2 =165 2 =256 2 =367 2 _ =498 2 =649 2 = = = =1 44
ALG 1B/ cdipaulo7 The square root of a non-perfect square. n is an irrational number. An Irrational number is a number that can not be written as a quotient of integers aka fraction. 2 = 1.414, this is a non-perfect square, it is irrational.
ALG 1B/ cdipaulo8 Radical Expressions A radical expression is any expression written with a radical. The radical acts as grouping symbol. What is under the radical is simplified first. Then take the root of what is underneath. Evaluating a radical expression is the same as evaluating any expression Write, replace and simplify.
ALG 1B/ cdipaulo9 Evaluate the radical… (b 2 - 4ac) When a =1, b = -2 and c = -3 Replace the variable with its value and evaluate. ((-2) 2 - 4(1)(-3)) = = 16 = 4
ALG 1B/ cdipaulo10 Square roots and TI You can evaluate a square root on your calculator and round the result if asked to do so. Here the + - means you have to evaluate twice, one for the positive square root and once for the negative square root. Our expression has two solutions:
ALG 1B/ cdipaulo11 Evaluate the Radical Remember the + - means you must evaluate twice. Once for the positive square root Once for the negative square root. 1.24, , ,.57
ALG 1B/ cdipaulo12 questions Summary : All positive real numbers have TWO square roots. –One positive –One negative –Written with a plus minus sign ±
ALG 1B/ cdipaulo13 Simplifying Roots Taking the square root undoes a square. 16 = ±4because 4 2 = 16 of a perfect square is rational, an integer. x 2 = 81the x = 9 and x = -9 or x = ±9 of non perfect square is irrational, written as a radical expression. y 2 = 3then y = ±3
ALG 1B/ cdipaulo15 The quadratic equation… ax 2 + bx + c = 0 –In standard form –The leading coefficient a 0 When b = 0 the equation becomes ax 2 + c = 0 Here you can solve for x by isolating the squared variable and using the square root.
ALG 1B/ cdipaulo16 When ax 2 + c = 0 x = 0 n = 0
ALG 1B/ cdipaulo17 The square root of a non perfect square When the answer to a squared number is not a perfect square, n 2 = 5 The answer will be given as a radical expression –more exact answer then using a decimal. Here: n = 5 AND - 5 Or n = ± 5
ALG 1B/ cdipaulo18 Solutions and the Quadratic NO Real Solution No real solution if the square of a number equals is negative. y 2 = -1 Here the square of a real number is NEVER negative! There is no real solution when: x 2 = d and d < 0 ONE Real Solution x 2 = 0 The 0 = 0 There is no other solution, zero is neither positive or negative. ONE real solution when x 2 = d and d = 0
ALG 1B/ cdipaulo19 Solutions and the Quadratic TWO Real Solutions All positive real numbers have 2 square roots. one positive square root one negative square root x = 0 x 2 = 81 x = ± 9 There are 2 real solutions when: x 2 = d and d > 0 Determine the solutions by solving the quadratic. x 2 = 81 y = 0 5n = 0 2x 2 - 2= 0
ALG 1B/ cdipaulo20 Quadratics and Real Life When a ball (or any object) is dropped the speed at which it falls continuously increases. Ignoring air resistance, the height of the ball h can be approximated by the falling object model: h = - 16 t 2 + s h is the height t is the time in seconds s is the initial height from where the ball is dropped. This model lets you look at the height of the object at anytime during its fall.
ALG 1B/ cdipaulo21 Falling Object Model Cat Ps most famous science teacher Mr. Tschachler is a contestant in an egg dropping contest. The goal is to create a container for the egg so it can be dropped from a height of 32 feet without breaking. Write a model for the eggs height, disregard air resistance. Use the model to determine how long it will take the egg to reach the ground. If h = -16t 2 + s then…
ALG 1B/ cdipaulo22 Algebraic Model for Mr. Tschachlers egg drop. h = -16t 2 + s What is the initial condition, where are we starting? 32 feet above ground. Replace the variable s with 32, this is the initial condition. Ground level is represented by 0, replace the height h with 0. The model for the egg drop is: 0 = -16t Solve for t, time. 0 = -16t = -16t 2 2 = t 2 ±2 As a decimal, t 1.4 seconds