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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Writing a Quadratic with Given Solutions

Not every quadratic equation can be solved by factoring, but the factoring method can be used (in reverse) to write a quadratic equation with any given solutions. For example, if the solutions to a quadratic equation are 5 and -3, we can reverse the steps in the factoring method as follows:

 x = 5 or x = -3 x - 5 = 0 or x + 3 = 0 (x - 5)(x + 3) = 0 Zero factor property x2 - 2x - 15 = 0 Multiply the factors.

This method will produce the equation even if the solutions are irrational or imaginary.

Example

Writing a quadratic given the solutions

Write a quadratic equation that has each given pair of solutions.

a) 4, -6

b)

c) -3i, 3i

Solution

a) Reverse the factoring method using solutions 4 and -6:

 x = 4 or x = -6 x - 4 = 0 or x + 6 = 0 (x - 4)(x + 6) = 0 Zero factor property x2 + 2x - 24 = 0 Multiply the factors.

b) Reverse the factoring method using solutions and :

 x = x + = 0 oror xx = = 0 (x + )(x ) x2 - 2 = 0= 0 Zero factor propertyMultiply the factors.

c) Reverse the factoring method using solutions -3i and 3i:

 x = -3ix + 3i = 0 oror xx - 3i = 3i= 0 (x + 3i)(x - 3i)x2 - 9i2 x2 + 9 = 0= 0 = 0 Zero factor propertyMultiply the factors Note: i2 = -1