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Chapter 1 Section 1.4

Prepared by Doron Shahar. Chapter 1 Section 1.4. Quadratic Equations. 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.

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Chapter 1 Section 1.4

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  1. Prepared by DoronShahar Chapter 1 Section 1.4 Quadratic Equations

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

  3. Prepared by DoronShahar FOIL and Factoring Factor FOIL First Outside Inside Last

  4. Prepared by DoronShahar 1.4.1 Solve by Factoring Starting Equation Factor Zero Product Property Solution

  5. Prepared by DoronShahar 1.4.2 Solve by Factoring Starting Equation Factor Zero Product Property Solution

  6. Prepared by DoronShahar Solve by Factoring Starting Equation FOIL Factor Zero Product Property Solution

  7. Prepared by DoronShahar 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

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

  9. Prepared by DoronShahar 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

  10. Prepared by DoronShahar 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

  11. Prepared by DoronShahar Example: Completing the square Starting Equation Add 8 to both sides Multiply both sides by 2 Desired form

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

  13. Prepared by DoronShahar 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

  14. Prepared by DoronShahar 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

  15. Prepared by DoronShahar 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

  16. Prepared by DoronShahar No solutions in quadratic equation Try solving 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. Take square root ¡PROBLEMA! You CANNOT take the square root of a negativenumber. Therefore, the equation has no solution.

  17. Prepared by DoronShahar 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

  18. Prepared by DoronShahar 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

  19. Prepared by DoronShahar Quadratic Formula Song Quadratic Formula Please sing along.

  20. Prepared by DoronShahar 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

  21. Prepared by DoronShahar Simplify your solution Simplify Solution

  22. Prepared by DoronShahar 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

  23. Prepared by DoronShahar Simplify your solution Simplify Solution

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

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

  26. Prepared by DoronShahar Discriminant

  27. Prepared by DoronShahar 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.

  28. Prepared by DoronShahar Simplifying expressions with

  29. Prepared by DoronShahar 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.

  30. Prepared by DoronShahar 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.

  31. Prepared by DoronShahar 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 hasB +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 willhave no variables in the denominators.

  32. Prepared by DoronShahar 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.

  33. Prepared by DoronShahar 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.

  34. Prepared by DoronShahar 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

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