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Paper 1 Algebra

Paper 1 Algebra. Leaving Certificate Helpdesk 20 th September 2012. General Content for Algebra. Simultaneous Equations Modulus Equations Inequalities The Nature of Roots of a Quadratic Equation Complex Numbers. Simultaneous Equations: Example 1. Solve the simultaneous equations:

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Paper 1 Algebra

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  1. Paper 1Algebra Leaving Certificate Helpdesk 20th September 2012

  2. General Content for Algebra • Simultaneous Equations • Modulus Equations • Inequalities • The Nature of Roots of a Quadratic Equation • Complex Numbers

  3. Simultaneous Equations: Example 1 Solve the simultaneous equations: _______________________________________ Step 1: Eliminate one of the variables.

  4. Step 2: Solve for either or using the following equations: Step 3: Solve for by subbing for in the equation:

  5. Step 4: Solve for using one of the original equations. We know and Answers:

  6. Simultaneous Equations in Three Variables Method: • Select one pair of equations and eliminate one of the variables. • Select another pair and eliminate the same variable. • Solve these two new equations simultaneously. • Use answers to find third variable.

  7. Simultaneous Equations: Example 2 Solve the simultaneous equations

  8. Simultaneous Equations: Example 3 2012 Paper 1 Q1(a)

  9. Rational Inequalities Method: Turn the rational inequality into a quadratic inequality by multiplying both sides by a positive expression. Example: Solve the inequality Note: multiplying both sides by a squared value ensures that the inequality sign is not affected.

  10. Complete all multiplication and tidy up the expression

  11. Solve the Quadratic to find the roots so that we can sketch the graph of the quadratic.

  12. Roots: When is ? Answer:

  13. Modulus Equations / Inequalities Solution: Square both sides

  14. Solve the quadratic to find the roots and sketch the curve: Complete all multiplication and tidy up the expression:

  15. Roots: Where is ? Answer: The inequality is true when

  16. The Nature of Roots of a Quadratic Example: 2009 Question 2 (b)(i)

  17. The Nature of Roots of a Quadratic Two real roots: Equal roots: Note: Roots are real if

  18. The Nature of Roots of a Quadratic Imaginary Roots:

  19. Quadratic Roots Example 1 The equation has equal roots. Find the possible values of k. Solution: Equal roots:

  20. Quadratic Roots Example 2 Sample Paper 2012 Paper 1 Q3

  21. If is a root then Conclusion: is a root of

  22. If is a root then is a factor of Solution: Divide into

  23. We now know: Solve to find final two roots Use

  24. Roots:

  25. The real root must be as we are told at the start that Thus are the imaginary roots Therefore

  26. Answer:

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