5.2 Solving Quadratic Equations by Factoring 5.3 Solving Quadratic Equations by Finding Square Roots

Objectives 1.Factor quadratic expressions and solve quadratic equations by factoring. 2.Find zeros of quadratic functions. Assignment: p 260 #24-30E, 36-40E, 66-74e Pg. 267 # 6-10, 12-17,60-68even, 71,72

Recall multiplying these binomials to get the standard form for the equation of a quadratic function: (x + 3)(x + 5) =+ 5x+3x+15 The “reverse” of this process is called factoring. Writing a trinomial as a product of two binomials is called factoring. (x + 3)(x + 5)

Factor Since the lead coefficient is 1, we need two numbers that multiply to –28 and add to –12. Factors of -28-1,281,-28-2,142,-14-4,74,-7 Sum of Factors Therefore:

Factor the expression: = (x-3)(x+7) Cannot be factored

Factoring a Trinomial when the lead coefficient is not 1. Factor: Multiply the coefficient of x 2 (a) by the constant term c to get a*c. Find two numbers whose sum is equal to the middle term and product equals a*c

Special Factoring Patterns you should remember: Pattern Name Pattern Example Difference of Two Squares Perfect Square Trinomial

Factor the quadratic expression:

A monomial is an expression that has only one term. As a first step to factoring, you should check to see whether the terms have a common monomial factor. Factor:

You can use factoring to solve certain quadratic equation. A quadratic equation in one variable can be written in the form where This is called the standard form of the equation: If this equation can be factored then we can use this zero product property. Zero Product Property Let A and B be real number or algebraic expressions. If AB = 0 the either A=0 or B=0

Solve: So, either (x+6)=0x = -6 Or (x – 3)=0x = 3 The solutions are –6 and 3. These solutions are also called zeros of the function Notice the zeros are the x-intercepts of the graph of the function.

Practice 1. x 2 -3x-4 =0 2. 5x 2 -13x+6=0 3. Find the zeros of the equation y = x 2 -4

Practice 1. x 2 -3x-4 =0 -1,4 2. 5x 2 -13x+6=0 3/5, 2 3. Write the equation in intercept form and find the zeros of the equation y = x 2 -4 (-2,2)

5.3 Solving Quadratic Equations by Finding Square Roots (p. 264)

Falling Objects! Use h = -16t 2 + h 0 Height of the object after it has fallen # of seconds after the object is dropped Object’s initial height

Example The tallest building in the USA is in Chicago, Illinois. It is 1450 ft. tall. How long would it take a penny to drop from the top of the building to the ground?

How would you solve the equation: x 2 = 4 (take the square root of each side!) * Remember, the square root of a positive # has 2 answers! (one + and one -)

Radical Radical sign Radicand

Properties of Square Roots (a>0 and b>0) 1.Product Property – 2.Quotient Property- Example: Example:

Examples

Rationalizing the Denominator You CANNOT leave a radical in the denominator of a fraction! No radicals in the basement!!!! (the numerator is OK) Just multiply the top & bottom of the fraction by the radical to “rationalize” the denominator.

More Examples! Can’t have a radical in the basement!

Solving Quadratic Equations 1.Solve x 2 = x 2 = x 2 = 2.Solve. 3(x-2) 2 = (x-2) 2 = 7

More Examples! 3.Solve. 4x 2 -6= x 2 = x 2 = Solve.

Assignment