# Quadratic Equations

**Definitions:** Quadratic equation:

(General form)

where a, b, c are constants.

**Note:** If a quadratic equation does not have a general form, we can always
rewrite it into a general

form.

E.g.,

** Solving Quadratic Equations
**

We will only consider real number solutions .

Always modify the equation in general form first. Then use one of two methods :

1. Factoring 2. The quadratic formula |

If the given equation is a rational
equation , check the values that make the denominator(s) zero so

that you can eliminate them from the solution.

** Factoring Method **

** Zero product property :** ab = 0, where a, b are real
a = 0 or b = 0 or
both.

Ex.1 (#10) Solve

Ex.2 Solve

Ex.3 (#16) Solve (x - 3)(1 - x) = 1

Ex.4 (p.142) Solve

**The Quadratic Formula **

** Square root property: **The solutions of x^{2} = C is

(Or one can think this as factoring

which gives
)

Ex.5 (#34) Solve

** Quadratic Formula :** If ax^{2} + bx + c = 0, where a ≠ 0, then

Sign of b ^{2} - 4ac |
Solutions |

Two distinct real solutions Exactly one real solution No real solutions |

Ex.6 (#22) Solve

Ex.7 (#24) Solve

**Applications**

**Note: **There may be solutions to a quadratic equation that are not the solutions
to a problem.

Ex.8 (#44) If the pro t from the sale of x units of a product is
what level(s) of

production will yield a pro t of $180?

Ex.9 (#48) A tennis ball is thrown into a swimming pool
from the top of a tall hotel. The height of the

ball from the pool is modeled by

feet,

where t is the time, in seconds, after the ball is thrown. How long after the
ball is thrown is it 4

feet above the pool?

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