Quadratic Formula

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b2 −4ac = 0

5x2 +6x +1=0

−6± 62 −4 ⋅5⋅12⋅5

=−6±4

10

=x

=x − .2,−1{ }

x

Plug intoQuadraticFormula

Simplify

Solve plusand minus forboth solutions

-1.0 -0.5-1.5 0.5

-0.5

0.5

x =− b ± b2 −4ac

2a

If

then

QuadraticFormula= ax2 +bx + c0

Discriminent

b2 −4ac < 0

b2 −4ac > 0Two real solutions when

One real solution when

Zero real solutions andTwo complex solutions

when

Rule

Example 2Example 1Two real solutionsZero real solutions and

Two complex solutions

x2 −4x +5=0

x =4± 42 −4 ⋅1⋅5

2⋅1

x =4± −4

2x =

4±2i2

x = 2+ i , 2− i{ }2.0

0.5

1.0

1.5

0.5 1.0 1.5 2.0 2.5 3.0

the two complexsolutions

no real solutions becausefunction does not cross x-axis