Algebra I: The Quadratic Equation

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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:

There are 2 solutions (roots) here.

Because they are such a mouthful one is called a and the other b.
It always works

(b) Sum and Product of the Roots

Remember as:

Remember it as:

(d) The roots satisfy their own equation
If you are told that something is a root of

then you can plonk it in. So if k is a root of

(e) The Discriminant
In the Magic the expression under the square root

discriminates between different types of roots.
If

there are 2 different real roots.
If

there are 2 equal real roots (this is the condition for equal roots).
Therefore if

there are real roots.
If

there are 2 complex (non-real) roots.

(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 1

If a and b are the roots of

evaluate
(i) a + b (ii) ab (iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

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,