Education LinksLeaving Cert
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The Quadratic Equation:
![]() For the Leaving Cert. you must know 6 things about this equation. (a) How to solve it: 2 methods (i) Factorization - quick but doesn't always work(ii) The Magic: ![]() Notes:
![]() Remember as: ![]() ![]() Remember as: ![]() If the roots are a and b the equation is ![]() Remember it as: ![]() (d) The roots satisfy their own equation If you are told that something is a root of ![]() ![]() (e) The Discriminant In the Magic the expression under the square root ![]() If ![]() If ![]() Therefore if ![]() If ![]() (f) Graphs of Quadratics The graphs of all quadratics are either Concave up (CUP) or Concave down (CAP) ![]() ![]() The roots a and b are the places where the curve crosses the x-axis. A number of different types of problems involving the quadratic are now examined. Type 1: Functions of a and b Example 1If a and b are the roots of![]() (i) a + b (ii) ab (iii) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Solution Type 2: Relationship between roots Given a relationship between roots a and b you can be asked to find a relationship between the co-efficients a, b, c. Let us consider some possibilities: (i) one root is double the other: a, 2a (ii) the roots add to 6: a, 6 - a (iii) the sum of the roots is zero: a, -a (iv) the roots are equal: a, a or b2 = 4ac (better) (v) the product of the roots is 3: a, ![]() (vi) one root is the reciprocal of the other: a, ![]() (vii) the roots are in the ratio 3:4: 3a, 4a Example 2If one root of![]() ![]() Solution Example 3If the roots of![]() ![]() Solution Type 3: New for Old/Two Quadratics Example 4If a and b are the roots of![]() ![]() Solution Example 5If a and b are the roots of![]() ![]() Solution |
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