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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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