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Algebra 2 – Chapter 5

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  1. Algebra 2 – Chapter 5 Quadratics

  2. Angry Birds! • Angry Birds follow a parabolic path. Your quest is to find the equation of the parabola to cause the angry bird to hit the pig. Use the given information and the quadratic regression method to find the equation of the curve. • All numbers are decimals – round to the thousandths place. • When you think your answer is correct, bring it to me to check it on the computer. Groups that successfully find all four equations will earn a prize. (woooo!)

  3. Homework: • page 245 (1-19, 33-37) odd

  4. 5-2 Properties of Parabolas

  5. maximum or minimum

  6. homework:

  7. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors? • Factor these expressions – Algebra 1 Review – Do you remember?

  8. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors? • Factoring is rewriting an expression as the product of its factors. • The greatest common factor (GCF) of the expression is a common factor of the term of the expression.

  9. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors? • When you factor a quadratic expression in the form ax2 + bx +c you are looking for a pair of factors that multiply to equal ac and add to equal b.

  10. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors?

  11. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors?

  12. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors?

  13. 5-4 Factoring Quadratic ExpressionsEQ: How do you reduce a quadratic expression into its linear factors? • Homework: page 268 (1-45) every other odd

  14. 5-4 solving Quadratic equations • Solving a quadratic equation means finding the values of the variable that make the equation true. • Usually, for a quadratic equation, there are two solutions. • There are several methods to solve quadratic equations: • Factoring • Finding Square Roots • Completing the Square • Using the Quadratic Formula

  15. 5-4 solving Quadratic equations • Solving by factoring requires: • Setting the equation equal to zero • Completely factoring the equation • Using the Zero-Product property to find the zeros. • Set each factor equal to zero and solve for the variable. This solution is called a zero of the equation because it makes the equation equal zero. ZERO PRODUCT PROPERTY: If ab = 0 then a = 0 or b = 0 Example: if then

  16. 5-4 solving Quadratic equations ZERO PRODUCT PROPERTY: If ab = 0 then a = 0 or b = 0 Example: if then • Standard Form for a Quadratic Equation:

  17. 5-4 solving Quadratic equations • Use factoring to solve the following equations:

  18. 5-4 solving Quadratic equations • Solving by finding Square Roots is used when there is no linear term. • Rewrite equation as • Isolate • Find square roots (remember, there are two!) • Example: • Try These:

  19. Simplifying Square Roots =

  20. Simplifying Square Roots • Break the number in the radical down to its prime factors – use a factor tree or repeated division. • 72 = 9 ∙ 8 = 3 ∙ 3 ∙ 4 ∙ 2 = 3 ∙ 3 ∙ 2 ∙ 2 ∙ 2 • Each pair of factors represents a single root that you can solve out of the radical = 6

  21. Simplifying Square Roots • Process is true for variables as well • Every pair of variables represents a single root variable =

  22. 5-6 Complex NumbersHow do you take the square root of a negative number? • Up until now, there was no way to deal with a root like this: • The letter i is defined as the square root of negative 1, and can be simplified out of a square root. • The numeral is rationalized in the usual way • =

  23. 5-6 Complex NumbersHow do you take the square root of a negative number?

  24. 5-6 Complex NumbersHow do you take the square root of a negative number? • Use the Complex Number Plane to represent a complex number geometrically. • Locate the real part of the complex number on the horizontal axis and the complex part on the vertical axis.

  25. 5-6 Complex NumbersHow do you take the square root of a negative number? • The absolute value of a complex number is its distance from the origin in the complex number plane. • You can find the absolute value by using the Pythagorean Theorem. • Find the absolute value:

  26. 5-6 Complex NumbersHow do you take the square root of a negative number? • When you add or subtract complex numbers you combine the real parts and imaginary parts separately. • When you multiply complex numbers you use the rules for multiplying binomials (FOIL) • Remember that i2 = -1

  27. 5-6 Complex NumbersHow do you take the square root of a negative number?

  28. 5-6 Complex NumbersHow do you take the square root of a negative number?

  29. 5-6 Complex NumbersHow do you take the square root of a negative number? • Write each answer in a + bi form

  30. 5-7 Completing the SquareUsing perfect squares to solve equations