Factoring Quadratic Expressions. Quadratic Equations What you’ll learn To factor polynomial expressions. To solve quadratic equations by factoring. To solve quadratic equations by graphing. Vocabulary Factoring, GCF, perfect square trinomial difference of two squares, zero of a function, zero product property.

Take a note What is a factoring? Factors of a given number are numbers that have a product equal to the given number. Factors of a given expressions that have a product equal to the given expression. Factoring is rewriting an expression as a product of its factors. GFC the greatest common factor of an expression is a common factor of the terms in the expression. Is the common factor with the greatest coefficients and the greatest exponents You can factor any expression that the GCF not equal to one.

Problem 1: Factoring when Answer A: Factors of 20 1,20 2,10 4,5 Sum of factors 21 12 9 Answer B: Factors of 72 -1,72 -2,36 -3,24 -4,18 -6,12 -8,9 Sum of factors 71 34 21 14 6 1 Answer C: First factor out -1 so Factors of 12 -1,-12 -2,-6 -3,-4 Sum of factors -13 -8 -7

Your turn What is the expression in factored form? Answers:

Problem 2 Finding common factors What is the expression factored from? Get GCF=3 and the variable with the least exponent The GCF=4 and then do the factorization Your turn What is the expression factored from?

Problem 3 Factoring when What is the expression of the factored form? I can not simplify the whole expression Then multiply a times c Then factor 1,24 2,12 3,8 4,6 Sum of factors 25 14 11 10 Rewrite the expression using b=3+8 Group them and find common factor

Your turn What is the expression in the factored form of? Can you factor this expression into a product of two binomials? Explain No, there is no factors of a and c whose product is 1 and whose sum is 1.

Your turn again What is the expression in a factored form of ? Another way to do it is by verifying that

Take a note When this happen you are factoring a perfect square trinomial. If ax²+bx+c is a perfect square trinomial, then ax² and c are squares of the binomial and thus are both positive. bx is twice the product of the terms of the binomial. If b is negative then the binomial terms have opposite signs. The expression a²-b²is the difference of two squares There is a pattern of factors(a-b)(a+b)

Your turn again What are the factors of?

Quadratic Equations Take a note Wherever the graph of a function f(x) intersects the x-axis, f(x)=0. A value of x for which f(x)=0 is a zero of the function. To find the zeros of a quadratic function y=ax²+bx+c Solve the related quadratic function by factoring and make the equation 0=ax²+bx+c You can solve some quadratic equations in standard form by factoring the quadratic expression using the zero product property. If ab=0 then a=0 or b=0

Problem 4: Solving a quadratic equation by factoring What are the solutions of the quadratic equation Factor the quadratic equation Use the zero product property The solutions are x=2 and x=3 If you are graphing this points will be (2,0) and (3,0)

Problem 5 385 ft. 74 ft. c)Domain: 0 ≤ x ≤ 385 Answer: The function f (x) = -0.002x² + 0.77x models the path of a baseball, where f (x) gives the height of the ball and x gives the distance from where it is hit in feet. a. How far does the ball travel before hitting the ground? b. How high does the ball go? c. What is a reasonable domain and range for such a function? 385 ft. 74 ft. c)Domain: 0 ≤ x ≤ 385 Range: 0 ≤ y ≤ 74 Answer: 395

c)domain: 0 ≤ x ≤ 21 and range: 0 ≤ x ≤ 3 Your turn From the time Mark Twain wrote The Celebrated Jumping Frog of Calaveras County in 1865, frog jumping competitions have been growing in popularity. The graph shows a function modeling the height of frog’s jump, where x is the distance, in feet, from the jump’s start. a)How far the frog jump? b) How high the frog jump? c)What is the reasonable domain and range for such a frog-jumping function? a) the frog jump about 20.34 ft. Answer b) y=-0.029(10.17)²+0.59(10.17)≈3.0 c)domain: 0 ≤ x ≤ 21 and range: 0 ≤ x ≤ 3

Classwork odd Homework even Text book pages 221-222 exercises 14-90 Text book pages 229-230 exercises 9-59