Quadratic Equations
Example
The solutions of a quadratic equation are 5 and
Work backwards to find a quadratic equation with these solutions.
Solution
Begin by writing the two solutions.


To clear the fraction in the second
solution, multiply both sides of the
second equation by 3.

x = 5 or 3x = 2 
Move the constant to the left side of
each equation. 
x  5 = 0 or 3x + 2 = 0 
Since each binomial is equal to 0,
their product is 0. 
(x  5)(3x + 2) = 0 
Multiply the binomials.
Combine like terms. 
3x^{2} + 2x  15x  10 = 0
3x^{2}  13x  10 = 0 
The quadratic equation 3x^{2}  13x  10 = 0
has the given solutions, 5 and
Note:
3x^{2}  13x  10 = 0
Any nonzero multiple of this equation
also has solutions 5 and
Here are some examples:
6x^{2}  26x  20 = 0
15x^{2}  65x  50 = 0
The quadratic formula states that the solutions of a quadratic equation,
ax^{2} + bx + c = 0, are
Letâ€™s see what happens when we combine the solutions first by addition
and then by multiplication:
â€¢ When we add the solutions,
the radicals are eliminated.


Add the numerators to
form one fraction. 

Combine like terms. The
square root terms add to
zero. 

Cancel the common
factor, 2. 

Thus, for a quadratic equation, ax^{2} + bx + c = 0, the sum of the solutions
is
â€¢ When we multiply the solutions, the radicals are eliminated.
Multiply the numerators and multiply the denominators.
Combine like terms.
Cancel the common factor, 4a.
Thus, for a quadratic equation, ax^{2} + bx + c = 0, the product of the solutions is
