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Quadratic Formula

The quadratic formula, first discovered by the Babylonians four thousand years ago, gives you a surefire way to solve quadratic equations of the form

0=a x 2 +bx+c .

Plugging in the values of a,b,andc , you will get the desired values of x .

x= b± b 2 4ac 2a

If the expression under the square root sign ( b 2 4ac , also called the discriminant) is negative, then there are no real solutions. (You need complex numbers to deal with this case properly. These are usually taught in Algebra 2 .)

If the discriminant is zero, there is only one solution. If the discriminant is positive, then the ± symbol means you get two answers.

Example 1:

Solve the quadratic equation.

x 2 x12=0

Here a=1,b=1,andc=12 . Substituting, we get:

x= ( 1 )± ( 1 ) 2 4( 1 )( 12 ) 2( 1 )

Simplify.

x= 1± 49 2

The discriminant is positive, so we have two solutions:

x= 8 2 and 6 2

x=4 and x=3

In this example, the discriminant was 49 , a perfect square, so we ended up with rational answers. Often, when using the quadratic formula, you end up with answers which still contain radicals.

Example 2:

Solve the quadratic equation.

3 x 2 +2x+1=0

Here a=3,b=2,andc=1 . Substituting, we get:

x= 2± 2 2 4( 3 )( 1 ) 2( 3 )

Simplify.

x= 2± 8 6

The discriminant is negative, so this equation has no real solutions.

 

Example 3:

Solve the quadratic equation.

x 2 4x+2=0

Here a=1,b=4,andc=2 . Substituting, we get:

x= ( 4 )± ( 4 ) 2 4( 1 )( 2 ) 2( 1 ) = 4± 168 2 = 4± 8 2

Simplify.

x= 4± 42 2 = 4±2 2 2 = 2( 2± 2 ) 2 =2± 2

The discriminant is positive but not a perfect square, so we have two real solutions:

x=2+ 2 and x=2 2