Quadratic Equations An Introduction

Slides:



Advertisements
Similar presentations
10-7 The Quadratic Formula
Advertisements

Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities
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 Objective: SWBAT solve quadratic equations and systems, and determine roots of a higher order polynomial.
EXAMPLE 4 Choose a solution method Tell what method you would use to solve the quadratic equation. Explain your choice(s). a. 10x 2 – 7 = 0 SOLUTION a.
Solving Quadratic Equations Using the Quadratic Formula MA.912.A.7.2 Solve quadratic equations over the real numbers by factoring and by using the quadratic.
The Quadratic Formula..
Lesson 9.8. Warm Up Evaluate for x = –2, y = 3, and z = – x 2 2. xyz 3. x 2 – yz4. y – xz 4 5. –x 6. z 2 – xy
3.5 Quadratic Equations OBJ:To solve a quadratic equation by factoring.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 1 Chapter 9 Quadratic Equations and Functions.
OBJ: To solve a quadratic equation by factoring
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)
Warm-ups Find each product. 1. (x + 2)(x + 7)2. (x – 11)(x + 5) 3. (x – 10) 2 Factor each polynomial. 4. x x x 2 + 2x – x 2.
“Parabola and Your Life”
“Parabola and Your Life” Quadratic Functions. The Zeros of Quadratic Functions The zeros of Quadratic Function f(x) = ax 2 + bx + c can be found by letting.
Get out Homework 9-6 Practice B Answers , , 2.48
10.6 Using the Quadratic Formula – Quadratic Formula Goals / “I can…”  Use the quadratic formula when solving quadratic equations  Choose an appropriate.
Chapter 10.7 Notes: Solve Quadratic Equations by the Quadratic Formula Goal: You will solve quadratic equations by using the Quadratic Formula.
Introduction Completing the square can be a long process, and not all quadratic expressions can be factored. Rather than completing the square or factoring,
10.5 Completing the Square – Completing the Square Goals / “I can…”  Solve quadratic equations by completing the square.
The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.
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.
Quadratic Equations An Introduction
Quadratic Equations: Factoring, Square Root Methods.
Solve by factoring. x² = - 4 – 5x 2,. Solve by factoring. n² = -30 – 11n -4 and -1.
©thevisualclassroom.com To solve equations of degree 2, we can use factoring or use the quadratic formula. For equations of higher degree, we can use the.
Chapter 4: Polynomial and Rational Functions. 4-2 Quadratic Equations For a quadratic equation in the form ax 2 + bx + c = 0 The quadratic Formula is.
PreCalculus Section P.1 Solving Equations. Equations and Solutions of Equations An equation that is true for every real number in the domain of the variable.
Completing the Square SPI Solve quadratic equations and systems, and determine roots of a higher order polynomial.
Quadratic Formula. Solve x 2 + 3x – 4 = 0 This quadratic happens to factor: x 2 + 3x – 4 = (x + 4)(x – 1) = 0 This quadratic happens to factor: x 2.
Warm Up 1.) What is the graph of the function y = -x 2 + 4x + 1?
Warm-Up Exercises EXAMPLE 1 Standardized Test Practice What are the solutions of 3x 2 + 5x = 8? –1 and – A 8 3 B –1 and 8 3 C 1 and – 8 3 D 1 and 8 3 SOLUTION.
Quadratic Equations An Introduction SPI Solve quadratic equations and systems, and determine roots of a higher order polynomial.
Bellwork 1)Write in standard form. 2) 3)A student is solving an equation by completing the square. Write the step in the solution that appears just before.
3.4 Chapter 3 Quadratic Equations. x 2 = 49 Solve the following Quadratic equations: 2x 2 – 8 = 40.
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.
SOLVE QUADRATIC EQUATIONS BY USING THE QUADRATIC FORMULA. USE THE DISCRIMINANT TO DETERMINE THE NUMBER AND TYPE OF ROOTS OF A QUADRATIC EQUATION. 5.6 The.
2.2 Solving Quadratic Equations Algebraically Quadratic Equation: Equation written in the form ax 2 + bx + c = 0 ( where a ≠ 0). Zero Product Property:
SOLVING QUADRATICS. Solving Quadratic Equations in Factored Form y = (x + 3)(x + 2) 0 = (x + 3)(x + 2) Ways to solve: y = x 2 + 5x + 6 x-intercepts, roots,
PreCalculus Section 1. 6 Solve quadratic equations by: a. Factoring b
Quadratic Equations An Introduction.
The Quadratic Formula..
Quadratic Equations An Introduction
4.6 Quadratic formula.
Solving Equations Graphically, Numerically, and Algebraically
The Quadratic Formula..
Solving Quadratic Equations
6.5 The Quadratic Formula and the Discriminant 2/13/07
The Quadratic Formula..
Quadratic Formula Solving for X Solving for quadratic equations.
The Quadratic Formula and the Discriminant
“Parabola and Your Life”
4.6 Quadratic formula.
The Quadratic Formula..
Solve x2 + 2x + 24 = 0 by completing the square.
9.3 Solving Quadratic Equations
The Quadratic Formula.
Objective Solve quadratic equations by graphing.
Review: Simplify.
Quadratic Equations.
Objective Solve quadratic equations by factoring..
Factor each of the following
Homework Check.
9.5 Factoring to Solve Quadratic Equations
Warm-up  .
Applying the Quadratic Formula
The Quadratic Formula..
The Quadratic Formula..
quadratic formula. If ax2 + bx + c = 0 then
Presentation transcript:

Quadratic Equations An Introduction SPI 3103.3.2 Solve quadratic equations and systems, and determine roots of a higher order polynomial.

Quadratic Equations are written in the form ax2 + bx + c = 0, where a ≠ 0.

Methods Used to Solve Quadratic Equations 1. Graphing 2. Factoring 3. Square Root Property 4. Completing the Square 5. Quadratic Formula

Why so many methods? - Some methods will not work for all equations. - Some equations are much easier to solve using a particular method. - Variety is the spice of life.

Graphing Graphing to solve quadratic equations does not always produce an accurate result. If the solutions to the quadratic equation are irrational or complex, there is no way to tell what the exact solutions are by looking at a graph. Graphing is very useful when solving contextual problems involving quadratic equations.

Graphing (Example 1) y = x2 – 4x – 5 Solutions are -1 and 5

Graphing (Example 2) y = x2 – 4x + 7 Solutions are

Graphing (Example 3) y = 3x2 + 7x – 1 Solutions are

Factoring Factoring is typically one of the easiest and quickest ways to solve quadratic equations; however, not all quadratic polynomials can be factored. This means that factoring will not work to solve many quadratic equations.

Factoring (Examples) Example 1 Example 2 x2 – 2x – 24 = 0 x2 – 8x + 11 is prime; therefore, another method must be used to solve this equation.

x2 = n or (x + c)2 = n Square Root Property This method is also relatively quick and easy; however, it only works for equations in which the quadratic polynomial is written in the following form. x2 = n or (x + c)2 = n

Square Root Property (Examples) Example 1 Example 2 x2 = 49 (x + 3)2 = 25 x = ± 7 x + 3 = ± 5 x + 3 = 5 x + 3 = –5 x = 2 x = –8 Example 3 x2 – 5x + 11 = 0 This equation is not written in the correct form to use this method.

Completing the Square This method will work to solve ALL quadratic equations; however, it is “messy” to solve quadratic equations by completing the square if a ≠ 1 and/or b is an odd number. Completing the square is a great choice for solving quadratic equations if a = 1 and b is an even number.

Completing the Square (Examples a ≠ 1, b is not even 3x2 – 5x + 2 = 0 Example 1 a = 1, b is even x2 – 6x + 13 = 0 x2 – 6x + 9 = –13 + 9 (x – 3)2 = –4 x – 3 = ± 2i x = 3 ± 2i OR x = 1 OR x = ⅔

Quadratic Formula This method will work to solve ALL quadratic equations; however, for many equations it takes longer than some of the methods discussed earlier. The quadratic formula is a good choice if the quadratic polynomial cannot be factored, the equation cannot be written as (x+c)2 = n, or a is not 1 and/or b is an odd number.

Quadratic Formula (Example) x2 – 8x – 17 = 0 a = 1 b = –8 c = –17