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Algebra Electronic Presentation Expert Voices F I N A L

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  • 1. Algebra Electronic Presentation: Expert Voices Pre Cal 30S January 22nd 2010 Emelda Iradukunda Haben Gabir Aruni Perera
  • 2. Algebra Table of Contents •  Absolute Values •  Solving for Roots by Completing the Square •  Solving for “p” using the Quadratic Formula •  Generating Equations with given Roots •  The Discriminant - The Nature of Roots •  Solving Rational Equations •  Solving Radical Equations
  • 3. Algebra Absolute Values
  • 4. Algebra Solving for Roots by Completeing the Square
  • 5. Algebra Solving for “p” using Quadratic Formula Equation: 4x - 10x² + 4 = 0 1) Substitute ‘p’ in place of x², therefore: p = x² => 4p² - 10p + 4 = 0 2) Solve for ‘p’ using the quadratic formula Quadratic Formula: 1x² + 1x + 1 = 0    a b c
  • 6. Algebra Solving for “p” using Quadratic Formula => -(-10) ± √(-10)² - 4(4)(4) 2(4) => 10 ± √ 100 - 64 8 => 10 ± √ 36 8 => 10 ± 6 8 Now we have 2 possible solutions: p1 = 16 = 2 p2 = 4 = 1  8  8 2 x1 = ± √2 x2 = √1 = 1 = ± √2 √2 √2 2
  • 7. Algebra Generating Equations with given Roots Formula for Equation: x² - (sum of roots)x + (product of roots) = 0 Roots: 4 + √6 and 4 - √6 1) Given the roots, we need to find the sum of the roots. 4 + √6 + 4 - √6 = 8 2) Given the roots, we need to find the product of the roots. 4 + √6 * 4 - √6 = 16 - 4√6 + 4√6 - 6 = 16 - 6 = 10
  • 8. Algebra Generating Equations with given Roots 3) Substitute the sum of the roots and product of the roots into the formula. x² - (sum of roots)x + (product of roots) = 0 => x² - 8x + 10 = 0
  • 9. Algebra The Discriminant - The Nature of Roots To find the value of the discriminant, we must use the formula: b² - 4ac Equation: 2x²- 2x - 6 = 0 1) Substitute the equation into the formula. => (-2)² - 4 (2)(-6) => 4 - (-48) = 52 2) Determine the nature of the roots of this value using the following rules:
  • 10. Algebra The Discriminant - The Nature of Roots b² - 4ac > 0 - there are two roots - if the value is a perfect square, the roots are rational - if the value is not a perfect square, the roots are irrational b² - 4ac = 0 - there is only one root - the function only crosses the x-axis at the vertex of the parabola b² - 4ac < 0 - the roots of the quadratic function are imaginary - the parabola does not cross the x-axis at any point
  • 11. Algebra The Discriminant - The Nature of Roots => 52 = 2 irrational roots 3) Find the exact value of the roots using this formula: - b ± √(value of discriminant) 2a => - (-1) ± √52 = 1 ± √52 2 (2) 4 Now we have 2 roots: r1 = 1 + √52 r2 = 1 - √52 4 4 4) Draw the quadratic on a graph.
  • 12. Algebra Solving Rational Equations Solving rational equations steps: Completely factorize the equation List all impossible values of x ( values that will make the denominator equal to 0 ) Get rid of any factors that cancel each other out Find the LCD and multiply it by both sides of equation ( this is done to get rid of the denominators )
  • 13. Algebra Solving Rational Equations Solve: The non-permissible values are: 2 (x can’t equal to 2) Nothing to factor, it is already factored. So now we multiply by the LCD, which in this case is x-2. 3x = 2x - 4 + 6 3x - 2x - 2 = 0 X-2=0 X=2
  • 14. Algebra Solving Rational Equations solve: Step1: The LCD is (x+1)(x-1) Non-permissible Values are x=1,-1 now we multiply both side by this 4x + 1 = 2x - 2 - x² - 1 x² + 4x + 4 = 0 (x+2)(x+2) = 0 x = -2
  • 15. Algebra Solving Radical Equations Radical equation is an equation that contains radicals or rational exponents. Solve by: •  Eliminating the radicals and obtain a linear or quadratic equation •  Solve the linear or quadratic using the method of quadratic and linear equations Important thing to remember when eliminating radicals: •  If a = b then a^n =b^n •  If you raise one side of an equation to a power , then you must keep the other side of the equation balanced by raising it to the same power
  • 16. Algebra Solving Radical Equations For Ex. =3 Square each side to get rid of square root sign (√x)² = (3)² x=9 solve: ³√x-5 = 0 Before raising both side of an equation to the nth power, you need to isolate the radical expression on one side of the equation ³√x = 5 (³√x)³ = (5)³ x = 125
  • 17. Algebra Equations Containing an Exponent x = 16 (x) ( ) = 16 x= x = 2³ x=8
  • 18. Algebra Practice Questions Solving for Roots by Completing the Square: x² + 2x + 3 = 0 Absolute Values 3x – 5 = 10 Solving for “p” using the Quadratic Formula: x - 5x² + 4 = 0 Generating Equations with given Roots: Given the roots 4 ± (5) ½, find the original quadratic equation.
  • 19. Algebra Practice Questions The Discriminant - The Nature of Roots x² - 8x + 16 = 0 Solving Rational Equations x = -2 x-3 Solving Radical Equations Simplify this radical equation: ( )
  • 20. Algebra Solutions to Practice Questions Solving for Roots by Completing the Square: y = (x + 1) ² + 2 Absolute Values x= 5, x = 5 3 Solving for “p” using the Quadratic Formula ±4, ±1 Generating Equations with given Roots x² - 8x + 21 = 0
  • 21. Algebra Solutions to Practice Questions The Discriminant - The Nature of Roots Discriminant = 0; one real root Solving Rational Equations x = 1, x = 2 Solving Radical Equations x=1 2
  • 22. Algebra Websites • http://www.youtube.com/user/yourteachermathhelp#p/u/ 574/GyCuj 1hx_zc • http://www.youtube.com/user/yourteachermathhelp#p/u/ 417/NxbLXwXXp7Y • http://www.youtube.com/watch? v=LY8VBsLf-4M&feature=related • http://www.youtube.com/user/yourteachermathhelp#p/u/ 427/FMfqBKfwkKc • http://www.purplemath.com/modules/absolute.htm • http://www.kutasoftware.com/FreeWorksheets/ Alg1Worksheets/Dividing%20Rational %20Expressions.pdf • http://www.kutasoftware.com/FreeWorksheets/ Alg1Worksheets/Simplifying%20Radicals.pdf