Solving Quadratics by Completing the Square & Quadratic Formula

Slides:

Advertisements

Similar presentations

TWO STEP EQUATIONS 1. SOLVE FOR X 2. DO THE ADDITION STEP FIRST

Advertisements

Angstrom Care 培苗社 Quadratic Equation II

5-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz

FACTORING Think Distributive property backwards Work down, Show all steps ax + ay = a(x + y)

The Discriminant Given a quadratic equation use the

Trig (Polar) Form of a Complex Number

Complex Numbers 3.3 Perform arithmetic operations on complex numbers

Factor P 16 8(8-5ab) 4(d² + 4) 3rs(2r – s) 15cd(1 + 2cd) 8(4a² + 3b²)

10-6 Solving Quadratic Equations by Factoring

25 seconds left…...

10-7 The Quadratic Formula

9-6 X-box Factoring ax 2 +bx+c. X-box Factoring This is a guaranteed method for factoring quadratic equations—no guessing necessary! We will learn how.

Jeff Bivin -- LZHS Polynomials and roots Jeffrey Bivin Lake Zurich High School Last Updated: October 26, 2009.

By: Jeffrey Bivin Lake Zurich High School Last Updated: October 11, 2005.

By: Jeffrey Bivin Lake Zurich High School Last Updated: October 9, 2007.

4.8 Use the Quadratic Formula and the Discriminant

4.7 Complete the Square.

EXAMPLE 4 Use the discriminant Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. a. x 2 – 8x +

4.8 Quadratic Formula and Discriminant

Graphing Parabolas Using the Vertex Axis of Symmetry & y-Intercept By: Jeffrey Bivin Lake Zurich High School Last Updated: October.

By: Jeffrey Bivin Lake Zurich High School Last Updated: October 30, 2006.

4.8: Quadratic Formula HW: worksheet

MATRICES Jeffrey Bivin Lake Zurich High School Last Updated: October 12, 2005.

Warmup 9-11 Solve the following equations by factoring. Show work! 1.x x - 80 = 0 2.Solve by using the quadratic formula: 4x 2 - 5x - 2 = 0 3.Solve.

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)

Quadratics Solving equations Using “Completing the Square”

Pre-Calculus Section 1.5 Equations Objectives: To solve quadratics by factoring, completing the square, and using the quadratic formula. To use the discriminant.

5.5 – The Quadratic formula Objectives: Use the quadratic formula to find real roots of quadratic equations. Use the roots of a quadratic equation to locate.

4.8 Do Now: practice of 4.7 The area of a rectangle is 50. If the width is x and the length is x Solve for x by completing the square.

Graphing Lines slope & y-intercept & x- & y- intercepts Jeffrey Bivin Lake Zurich High School Last Updated: September 6, 2007.

Section 4.7 – The Quadratic Formula Students will be able to: To solve equations using the Quadratic Formula To determine the number of solutions by using.

Warm Up #4 1. Write 15x2 + 6x = 14x2 – 12 in standard form. ANSWER

More about Quadratic Equations November 16, 2009.

Systems of Equations Gaussian Elimination & Row Reduced Echelon Form by Jeffrey Bivin Lake Zurich High School Last Updated: October.

The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.

Jeff Bivin -- LZHS Last Updated: April 7, 2011 By: Jeffrey Bivin Lake Zurich High School

Graphing Parabolas Using the Vertex Axis of Symmetry & y-Intercept By: Jeffrey Bivin Lake Zurich High School

1.3 Quadratic Equations College Algebra: Equations and Inequalities.

Jeff Bivin -- LZHS Quadratic Equations. Jeff Bivin -- LZHS Convert to Standard Form f(x) = 5x x + 46 f(x) = 5(x 2 - 8x + (-4) 2 ) f(x)

Notes Over 5.6 Quadratic Formula

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.

Matrix Working with Scalars by Jeffrey Bivin Lake Zurich High School Last Updated: October 11, 2005.

CHAPTER 10 LESSON OBJECTIVES. Objectives 10.1 Students will be able to: Identify quadratic functions and determine whether they have a minimum or maximum.

Warm Up  1.) Write 15x 2 + 6x = 14x in standard form. (ax 2 + bx + c = 0)  2.) Evaluate b 2 – 4ac when a = 3, b = -6, and c = 5.

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.

How do I solve quadratic equations using Quadratic Formula?

Do Now Use the standard form of a quadratic equation to find the a, b and c of each equation. ax2 + bx + c = 0 x2 – 6x + 10 = 0 2x2 + 3x + 4 = 0 x2 –

4.6 Quadratic formula.

The Quadratic Formula..

Introductory Algebra Glossary

Section 5-3: X-intercepts and the Quadratic Formula

Solving Quadratics by Completing the Square & Quadratic Formula

4.6 Quadratic formula.

5.6 The Quadratic Formula and the Discriminant

4.8 The Quadratic Formula and the Discriminant

Unit 7 Day 4 the Quadratic Formula.

The Quadratic Formula.

E) Quadratic Formula & Discriminant

Lake Zurich High School

Quadratic Formula & the Discriminant

Review: Simplify.

Lake Zurich High School

Warm Up #4 1. Write 15x2 + 6x = 14x2 – 12 in standard form. ANSWER

Using the Quadratic Formula to Solve Quadratic Equations

The Quadratic Formula..

QUADRATIC FUNCTION PARABOLA.

The Quadratic Formula..

Presentation transcript:

Solving Quadratics by Completing the Square & Quadratic Formula By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: October 24, 2007

X2 + 6x + 9 x 1 1 1 1 1 1 x x + 3 Now, complete the square 1 + 9 1 1 Jeff Bivin -- LZHS

X2 + 4x + 4 4 x 1 1 1 1 x x + 2 Now, complete the square 1 1 x + 2 Jeff Bivin -- LZHS

X2 + 5x + 25/4 x 1 1 .5 1 1 1 x x + 5/2 Now, complete the square 1 Jeff Bivin -- LZHS

X2 - 6x + 9 x 1 1 1 1 1 1 1 1 1 x - 3 x 1 + 9 1 1 turn 1 square over turn 2 squares over turn 2 squares over turn 3 squares over Jeff Bivin -- LZHS

Solve by Completing the Square x2 + 10x + 8 = 0 (x2 + 10x ) = -8 (x2 + 10x + (5)2) = -8 + 25 (5)2 = 25 (x + 5)2 = 17 Jeff Bivin -- LZHS

Solve by Completing the Square 3x2 + 24x + 12 = 0 3 x2 + 8x + 4 = 0 (x2 + 8x ) = -4 (x2 + 8x + (4)2) = -4 + 16 (4)2 = 16 (x + 4)2 = 12 Jeff Bivin -- LZHS

Solve by Completing the Square 2x2 + 5x - 12 = 0 2 Jeff Bivin -- LZHS

Solve by Completing the Square 2x2 - 12x - 11 = 0 2 Jeff Bivin -- LZHS

Solve by Completing the Square -5x2 + 12x + 19 = 0 -5 Jeff Bivin -- LZHS

Solve by Completing the Square 5x2 - 30x + 45 = 0 5 Jeff Bivin -- LZHS

Solve by Completing the Square 5x2 - 30x + 75 = 0 5 Jeff Bivin -- LZHS

Convert to vertex form y = x2 + 10x + 8 y - 8 = (x2 + 10x ) (5)2 = 25 y - 8 + 25 = (x2 + 10x + (5)2) y + 17 = (x2 + 10x + (5)2) y + 17 = (x + 5)2 - 17 x + 5 = 0 Axis of symmetry: x = -5 Vertex: (-5, -17) Jeff Bivin -- LZHS

Convert to vertex form y = 5x2 - 30x + 46 y - 46 = 5(x2 - 6x ) 5(-3)2 = 45 y - 46 + 45 = 5(x2 - 6x + (-3)2) y - 1 = 5(x2 - 6x + (-3)2) y - 1 = 5(x - 3)2 + 1 x - 3 = 0 Axis of symmetry: x = 3 Vertex: (3, 1) Jeff Bivin -- LZHS

Solve by Completing the Square ax2 + bx + c = 0 a The Quadratic Formula Jeff Bivin -- LZHS

Solve using the Quadratic Formula 3x2 + 7x - 4 = 0 a = 3 b = 7 c = -4 Jeff Bivin -- LZHS

Solve using the Quadratic Formula 6x2 + 9x + 2 = 0 a = 6 b = 9 c = 2 Jeff Bivin -- LZHS

Solve using the Quadratic Formula 5x2 - 8x + 1 = 0 a = 5 b = -8 c = 1 Jeff Bivin -- LZHS

Solve using the Quadratic Formula 6x2 - 17x - 14 = 0 a = 6 b = -17 c = -14 Jeff Bivin -- LZHS

Solve using the Quadratic Formula x2 + 6x + 9 = 0 a = 1 b = 6 c = 9 Jeff Bivin -- LZHS

Solve using the Quadratic Formula 3x2 + 7x + 5 = 0 a = 3 b = 7 c = 5 Two Imaginary Solutions Jeff Bivin -- LZHS

Why do some quadratic equations have 2 real solutions some have 1 real solution and some have two imaginary solutions?

Now Consider Discriminant ax2 + bx + c = 0 89 -11 If discriminant > 0, then 2 real solutions If discriminant = 0, then 1 real solution If discriminant < 0, then 2 imaginary solutions Jeff Bivin -- LZHS

That's All Folks Jeff Bivin -- LZHS