Factoring Quadratic Functions if a ≠ 1 (Section 3.5)

Slides:



Advertisements
Similar presentations
Finding Complex Roots of Quadratics
Advertisements

AC Method of factoring ax2 + bx +c
The Quadratic Formula for solving equations in the form
solution If a quadratic equation is in the form ax 2 + c = 0, no bx term, then it is easier to solve the equation by finding the square roots. Solve.
Solving Quadratic Equations Algebraically Lesson 2.2.
6 – 4: Factoring and Solving Polynomial Equations (Day 1)
Solving Quadratic Equations by Completing the Square
3.5 Quadratic Equations OBJ:To solve a quadratic equation by factoring.
Copyright © Cengage Learning. All rights reserved.
Solving Quadratic Equations by the Quadratic Formula
Section 10.5 – Page 506 Objectives Use the quadratic formula to find solutions to quadratic equations. Use the quadratic formula to find the zeros of a.
OBJ: To solve a quadratic equation by factoring
Algebra 1B Chapter 9 Solving Quadratic Equations By Graphing.
The Quadratic Formula. What does the Quadratic Formula Do ? The Quadratic formula allows you to find the roots of a quadratic equation (if they exist)
6.6 Quadratic Equations. Multiplying Binomials A binomial has 2 terms Examples: x + 3, 3x – 5, x 2 + 2y 2, a – 10b To multiply binomials use the FOIL.
Copyright © Cengage Learning. All rights reserved. Factoring Polynomials and Solving Equations by Factoring 5.
Solving Quadratic Equations. Solving by Factoring.
2.6 Solving Quadratic Equations with Complex Roots 11/9/2012.
The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.
Holt Algebra Solving Quadratic Equations by Graphing and Factoring A trinomial (an expression with 3 terms) in standard form (ax 2 +bx + c) can be.
MM2A3 Students will analyze quadratic functions in the forms f(x) = ax 2 +bx + c and f(x) = a(x – h) 2 = k. MM2A4b Find real and complex solutions of equations.
1.3 Quadratic Equations College Algebra: Equations and Inequalities.
Solving Quadratic Equations. Factor: x² - 4x - 21 x² -21 a*c = -21 b = -4 x + = -21 = x 3x3x x 3 (GCF) x-7 (x – 7)(x + 3)
Completing the Square SPI Solve quadratic equations and systems, and determine roots of a higher order polynomial.
Ch 10: Polynomials G) Completing the Square Objective: To solve quadratic equations by completing the square.
Ch. 6.4 Solving Polynomial Equations. Sum and Difference of Cubes.
Ch 10: Polynomials G) Completing the Square Objective: To solve quadratic equations by completing the square.
Factoring Trinomials By Grouping Method Factoring 5/17/20121Medina.
Chapter 5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-1 Factoring.
Lesson 6.5: The Quadratic Formula and the Discriminant, pg. 313 Goals: To solve quadratic equations by using the Quadratic Formula. To use the discriminant.
1.2 Quadratic Equations. Quadratic Equation A quadratic equation is an equation equivalent to one of the form ax² + bx + c = 0 where a, b, and c are real.
5x(x 2 – 4) (y 2 + 4)(y 2 – 4) (9 + d 2 )(9 – d 2 )
Algebra 1 Section 10.5 Factor quadratic trinomials and solve quadratic equations A quadratic trinomial takes the form: ax 2 + bx + c Example: (x+3)(x+4)
Graphing Quadratic Functions Solving by: Factoring
4.6 Quadratic formula.
Using the Quadratic Formula to Find Solutions
Chapter 4 Quadratic Equations
Splash Screen.
Quadratic Equations P.7.
Warm-up 1. Solve the following quadratic equation by Completing the Square: 2x2 - 20x + 16 = 0 2. Convert the following quadratic equation to vertex.
The Quadratic Formula..
Objectives Solve quadratic equations by factoring.
5.3 Factoring Quadratics.
Factoring Polynomials
Solve a quadratic equation
Section 5-3: X-intercepts and the Quadratic Formula
4.6 Quadratic formula.
Solving Quadratic Equations by the Quadratic Formula
The Quadratic Formula..
The Quadratic Formula.
5.9 The Quadratic Formula 12/11/2013.
Section 9.2 Using the Square Root Property and Completing the Square to Find Solutions.
3/3/ Week Monday: Review and take quiz on: Simplify in
The Square Root Property and Completing the Square
FOIL: Trinomial Factoring with lead coefficient of one
Review: 6.5b Mini-Quiz 1. Solve: 9x2 – 100 = 0.
Quadratic Equations and Functions
Review: Simplify.
You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Functions have zeros or x-intercepts. Equations.
Factoring ax2 + bx + c CA 11.0.
4.5: Completing the square
Warm Up #4 1. Write 15x2 + 6x = 14x2 – 12 in standard form. ANSWER
Solve quadratic equations using the: QUADRATIC FORMULA
Quadratic Formula & Discriminant
Applying the Quadratic Formula
The Quadratic Formula..
The Quadratic Formula..
Ch 10: Polynomials G) Completing the Square
Presentation transcript:

Factoring Quadratic Functions if a ≠ 1 (Section 3.5) MM2A4b. Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Solving Polynomial Equations If the problem is a trinomial of the form: ax2 + bx + c = 0, and a ≠ 1, we must factor the trinomial by finding the factors of a * c that add to b

Basic Steps Put equation in standard form – make a > 0 Factor out the GCF Determine a, b, and c Find the factors of a times c that add to b Rewrite the equation replacing the b term with the factors found above Factor by grouping Use the zero product rule to find the solutions Verify by substituting solutions into the original equation

Emphasize the “cross” idea and logic: If ac > 0: both factors must be positive or negative. If b > 0, they are both positive If b < 0, they are both negative If ac < 0: One factor is positive and one is negative If b > 0, the larger one is positive If b < 0, the larger one is negative

Product (ac) Factor 1 Factor 2 Sum (b)

Solving Polynomial Equations Example: find the zeros of 6x2 + x – 12 = 0 ac = -72 and the factors of -72 are: 2 and -36, 3 and -24, 4 and -18, 6 and -12, 9 and -8, -2 and 36, -3 and 24, -4 and 18, -6 and 12, -9 and 8 The two that add to 1 is 9 and -8, so we replace x with 9x and -8x and factor by grouping.

Product (ac = -72) Factor 2 (-8) Factor 1 (+9) Sum (b = 1)

Continuing: Example: find the zeros of 6x2 + x – 12 = 0 6x2 + 9x – 8x – 12 = 0 3x(2x + 3) – 4(2x + 3) = 0 (2x + 3)(3x – 4) = 0 x = -3/2 or x = 4/3

Summary Find the factors of “ac” that add to “b” If ac > 0: both factors must be positive or negative. If b > 0, they are both positive If b < 0, they are both negative If ac < 0: One factor is positive and one is negative If b > 0, the larger one is positive If b < 0, the larger one is negative