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### Forming quadratic equations from given zeros

A quadratic equation can be formed by its solutions. For example, if 2 and 3 are the solutions of an unknown quadratic equation, the quadratic equation can be formed as follows:

Sum of zeros, S = 2 + 3 = 5
Product of zeros, P = 2 * 3 = 6
Substitute the value of S and P in the following equation:
x^2 - Sx + P = 0
x^2 - 5x + 6 = 0
The quadratic equation x^2 - 5x + 6 = 0 has the two zeros, 2 and 3. You should check the solutions of the quadratic equation to ensure that you have got the correct equation. In order to do so, either apply the quadratic formula,
x = [ -(-5) +/- sqrt((-5)^2 - 4(1)(6)) ] / [ 2(1) ]
x = [ +5 +/- sqrt(25 - 24) ]  [ 2 ]
x = [ 5 +/- 1 ] / 2
x =  (5 + 1)/2 or x = (5 - 1)/2
x = 3 or x = 2, thus the zeros are correct.
or solve the quadratic equation by factoring it (if possible),
x^2 - 5x + 6 = 0
x^2 - 2x - 3x + 6 = 0
x(x - 2) - 3(x - 2) = 0
(x - 2)(x - 3) = 0
Either x - 2 = 0 or x - 3 = 0
x = 2 or x = 3, which are the original zeros.