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Quadratic Equations

Consider the equation

x2 - +25 = 0

This equation is an example of a quadratic equation. The real numbers +5 and -5 both satisfy the above open sentence:

The solution set of the given equation consists of simply the real numbers +5 and -5. We say that +5 and -5 are the two roots of the given quadratic equation.

When we are trying to find the roots of a quadratic equation, we often make use of an important principle of real numbers. This principle, which is derived from the basic principles listed in the preceding section, states that the product of a pair of real numbers is 0 if and only if one of the numbers is 0. That is,

x y xy = 0

if and only if

x = 0 or y = 0





Let us see how we may use this principle to find the roots of the quadratic equation

x2 - +16 = 0

We first notice that the expression

x2 - +16

may be factored:

x2 - +16 = x2 - +42 = (x - +4) - (x + +4)

Thus, we may transform the given equation

x2 - +16 = 0

into the equivalent equation

(x - +4) - (x + +4) = 0

A real number satisfies the last equation if and only if it satisfies the given equation.





We know that

(x - +4) - (x + +4) = 0

if and only if

x - +4 = 0 or x + +4 = 0

Thus, a real number satisfies the equation

(x - +4) - (x + +4) = 0

If and only if the real number satisfies one of the following equations:

x - +4 = 0 or x + +4 = 0

The solution set of the equation

x - +4 = 0

consists of simply the number +4, since

The solution set of the equation

x + +4 = 0

consists of simply the number -4, since

Thus, the solution set of the equation

(x - +4) - (x + +4) = 0

consists of the numbers +4 and -4. Hence the roots of the given quadratic equation

x2 - +16 = 0

are the numbers +4 and -4. We may check this result:





Here is another example that shows how factoring aids us in finding the roots of a quadratic equation:

Example: Find the roots of the equation

x2 - +3x = 0

Solution: The expression x2 - 3x may be factored

x2 - +3x = (x - +3) - x

Hence the given equation may be transformed into the equivalent equation

(x - +3) - x = 0

A real number satisfies this equation if and only if it satisfies one of the following equations:

x - +3 = 0 or x = 0

The only real number that satisfies the equation

x - +3 = 0

is +3, and the only real number that satisfies the equation

x = 0

is 0. Thus, the roots of the given equation are +3 and 0.



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