Prepared by Doron Shahar Chapter 1 Section 1.4 Quadratic Equations

Warm-up: page 15 The zero product property says that if , Prepared by Doron Shahar Warm-up: page 15 A quadratic equation is an equation that can be written in the form _____________ where a, b, and c are constants and a ≠ 0. The zero product property says that if , then either ________ or ________.

FOIL and Factoring Factor FOIL First Outside Inside Last Prepared by Doron Shahar FOIL and Factoring FOIL Factor First Outside Inside Last

1.4.1 Solve by Factoring Starting Equation Factor Prepared by Doron Shahar 1.4.1 Solve by Factoring Starting Equation Factor Zero Product Property Solution

1.4.2 Solve by Factoring Starting Equation Factor Prepared by Doron Shahar 1.4.2 Solve by Factoring Starting Equation Factor Zero Product Property Solution

Solve by Factoring Starting Equation FOIL Factor Zero Product Property Prepared by Doron Shahar Solve by Factoring Starting Equation FOIL Factor Zero Product Property Solution

Intro to Completing the square Prepared by Doron Shahar Intro to Completing the square Starting Equation Take square root of both sides of the equation Place ± on the right side of the equation Solution

Intro to Completing the square Prepared by Doron Shahar Intro to Completing the square Starting Equation Take square root Insert ± on right side Group like terms Solution

Goal of Completing the square Prepared by Doron Shahar Goal of Completing the square The goal of completing the squares is to get a quadratic equation into the following form: Starting Equation eg. Take square root Insert ± on right side Group like terms Solution

1.4.5 Completing the square Starting Equation Add 3 to both sides Prepared by Doron Shahar 1.4.5 Completing the square The goal of completing the squares is to get a quadratic equation into the following form: Starting Equation Add 3 to both sides Multiply both sides by 2 Desired form

Example: Completing the square Prepared by Doron Shahar Example: Completing the square Starting Equation Add 8 to both sides Multiply both sides by 2 Desired form

Completing the square Equation from previous slide Take square root Prepared by Doron Shahar Completing the square Equation from previous slide Take square root Insert ± on right side Group like terms Solution

1.4.2 General method of Completing the square Prepared by Doron Shahar 1.4.2 General method of Completing the square Starting Equation Add 9 to both sides Add (−8/2)2=16 to both sides Factor left side

1.4.4 General method of Completing the square Prepared by Doron Shahar 1.4.4 General method of Completing the square Starting Equation Divide both sides by 3 Add ((−2/3)/2)2=1/9 to both sides Factor left side

1.4.3 General method of Completing the square Prepared by Doron Shahar 1.4.3 General method of Completing the square Starting Equation Subtract 22 from both sides Add (−6/2)2=9 to both sides Factor left side

No solutions in quadratic equation Prepared by Doron Shahar No solutions in quadratic equation Try solving Take square root ¡PROBLEMA! You CANNOT take the square root of a negative number. Therefore, the equation has no solution. Solving by factoring works only if the equation has a solution. Completing the square always works, and can be used to determine whether a quadratic equation has a solution.

General method of Completing the square Prepared by Doron Shahar General method of Completing the square Starting Equation Subtract c from both sides Divide both sides by a Add ((b/a)/2)2=(b/2a)2 to both sides Factor left side

Quadratic Formula Quadratic Formula Starting Equation Solution Prepared by Doron Shahar Quadratic Formula If we solve for in the previous equation, , we get an equation called the quadratic formula. Quadratic Formula The quadratic formula gives us the solutions to every quadratic equation. Starting Equation Solution

Quadratic Formula Song Prepared by Doron Shahar Quadratic Formula Song Quadratic Formula Please sing along.

Using the quadratic formula Prepared by Doron Shahar Using the quadratic formula Starting Equation Plug 9 in for a, 6 for b, and 1 for c in the quadratic formula. Quadratic Formula Solution

Simplify your solution Prepared by Doron Shahar Simplify your solution Simplify Solution

1.4.1 Using the quadratic formula Prepared by Doron Shahar 1.4.1 Using the quadratic formula 1 Starting Equation Plug 1 in for a, 9 for b, and 14 for c in the quadratic formula. Quadratic Formula Solution

Simplify your solution Prepared by Doron Shahar Simplify your solution Simplify Solution

1.4.3 Using the quadratic formula Prepared by Doron Shahar 1.4.3 Using the quadratic formula +( ) Starting Equation 1 Plug 1 in for a, −6 for b, and 22 for c in the quadratic formula. Quadratic Formula Solution

Simplify your solution Prepared by Doron Shahar Simplify your solution Simplify You cannot take the square root of a negative number. Therefore, there is no solution. No Solution

Discriminant Equation Discriminant # of Solutions Prepared by Doron Shahar Discriminant Equation Discriminant # of Solutions

Prepared by Doron Shahar Calculator Put the Quadratic Formula program on your calculator. Instructions are in the back of the class notes. OR You can come in to office hours to have me load the program onto your calculator. Warning! The calculator will not always give you exact answers.

Simplifying expressions with Prepared by Doron Shahar Simplifying expressions with

Quadratic equations with Decimals Prepared by Doron Shahar Quadratic equations with Decimals If a quadratic equation has decimals, it is easiest to simply use the quadratic formula. If you want, you can multiply both sides of the equation by a power of 10 (i.e., 10, 100, 1000, etc) to get rid of the decimals. This can make it easier to simplify the answer if you are evaluating the quadratic formula without a calculator.

Quadratic equations with Fractions Prepared by Doron Shahar Quadratic equations with Fractions If a quadratic equation has fractions (and does not factor), it is often easy to simply use the quadratic formula. If you want, you can multiply both sides of the equation by the least common denominator to get rid of the fractions. This can make it easier to simplify the answer. If a quadratic equation has fractions (and factors), it is often easier to factor after having gotten rid of the fractions.

Variables in the denominators Prepared by Doron Shahar Variables in the denominators If an equation has variables in the denominator, it is NOT a quadratic equation. Such equations, however, can lead to linear equations. We treat such equations like those with fractions. That is, we multiply both sides of the equation by a common denominator to get rid of the variables in the dominators. Ideally, we should multiply by the least common denominator. Example: If our problem has B +16 in the denominator of one term, and B in the denominator of another term, we multiply both sides of the equation by B(B+16). After the multiplication, the terms will have no variables in the denominators.

Variables in the denominators Prepared by Doron Shahar Variables in the denominators Starting Equation Multiply both sides by B(B+16) Distribute the B(B+16) The problem is now in a form you can solve.

Prepared by Doron Shahar Systems of equations There are two methods for solving systems of equations: Substitution and Elimination. Both work by combining the equations into a single equation with one variable. And sometimes the resulting equation leads to a quadratic equation. We will only review substitution, because elimination is not a common method when working with systems of equations that lead to quadratic equations.

Substitution Starting Equations Substitute B+16 for J Prepared by Doron Shahar Substitution Starting Equations Substitute B+16 for J in the second equation Solution for B Plug in 24 and −10 for B in the first equation to get the solution for J