1. Quadratic Equations by Dr. Terri
5/25/99
Quadratic Equations by Dr. Terri

2. Quadratic Equations by Dr. Terri
A quadratic equation is of the form axn + bx + c = 0,
where a, b, and c are real numbers and n > 1.
Quadratic equations can be solved by
Factoring
Completing the square
Using the quadratic formula
5/25/99
Quadratic Equations by Dr. Terri

3. Quadratic Equations by Dr. Terri
Factor to Solve for x
Factor ax2 + bx + c = 0 by trial and error
a, b, and c are real numbers
Set both factors equal to zero to solve for x
Example:
2x2 + 3x - 2 = 0 factors as (2x - 1)(x + 2) = 0
The equation is true when either factor equals zero
2x - 1 = x + 2 = 0
2x = x = - 2
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Quadratic Equations by Dr. Terri

4. Quadratic Equations by Dr. Terri
Completing the Square
Rewrite ax2 + bx + c = 0 in the form a(x - h)2 + k = 0
Example: 2x2 + 3x - 2 = 0
Group the first and second degree terms together
(2x2 + 3x) - 2 = 0
Factor out the coefficient of the second degree term (a)
2(x x) - 2 = 0
Add the constant, c, to both sides
2(x x) = 2
(continued on next slide)
5/25/99
Quadratic Equations by Dr. Terri

5. Completing the Square page 2
2(x x) = 2
Take half the resulting coefficient of the first degree term ( )
( ) · ( ) =
Square the result
( )2 =
Add in and subtract back out this new amount inside the parentheses
2(x x ) = 2
(continued on next slide) 3 2 3 2 1 2 3 2 3 4 3 4 9 16 3 2 9 16 9 16
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Quadratic Equations by Dr. Terri

6. Completing the Square page 3
Determine the value of the last term inside the parentheses
2(- ) = (multiply by the outside factor)
Remove this last term from inside the parentheses
2(x x ) = 2
Bring this value to the other side of the equation and reduce
2(x x ) = =
If you have trouble with fractions or reducing, review those lessons.
(continued on next slide) 9 16 18 16 3 2 9 16 18 16 3 2 9 16 18 16 25 8
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Quadratic Equations by Dr. Terri

7. Completing the Square page 4
2(x x ) =
Express the quantity inside the parentheses as the square of a binomial
2(x + )2 = (trial and error factoring)
Divide both sides of the equation by the coefficient of the squared term
(x + )2 = =
Take the square root of both sides of the equation
x = 
(continued on next slide) 3 2 9 16 25 8 3 4 25 8 . 2 3 4 25 8 1 2 25 16 3 4 5 4
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Quadratic Equations by Dr. Terri

8. Quadratic Equations by Dr. Terri
Completing the Square page 5
x = 
Solve for x
x + = and x = -
x = x = -
= = - 2 3 4 5 4 3 4 5 4 3 4 5 4 2 4 8 4 1 2
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Quadratic Equations by Dr. Terri

9. Quadratic Formula to Solve for x
The equation ax2 + bx + c = 0 can be solved by
substituting values for a, b, and c into the formula
x = -b  b2 - 4ac 2a
This formula is known as the quadratic formula.
The quantity under the radical sign is known as the
discriminant, and helps determine the nature of the
resulting values for x.
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Quadratic Equations by Dr. Terri

10. Information from the Discriminant
Given an equation of the form ax2 + bx + c = 0
where b2 - 4ac (the discriminant) has been calculated,
the nature of the solutions are as follows:
Value of Discriminant Nature of Solutions
positive real
zero real
negative 2 non-real
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Quadratic Equations by Dr. Terri

11. Quadratic Equations by Dr. Terri
Complex numbers
Quadratic Equations may have non-real complex solutions
a + bi represents a complex number, where a and b are real numbers
If b = 0, the complex number is a real number
If b  0, the complex number is then a non-real number
i, by definition, is the square root of -1. It is an imaginary number.
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Quadratic Equations by Dr. Terri

12. Quadratic Equations by Dr. Terri
Complex Numbers page 2
Example:
x2 - 6x + 13 = 0
This equation is not easily factored. It can be
solved by completing the square or by using the
quadratic formula.
By using the quadratic formula, the solutions are:
a = 1, b = -6, c = 13
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Quadratic Equations by Dr. Terri

13. Solving Quadratics: Exercises
Solve -4x2 + x + 14 = by factoring
( 4x + 7) (-x + 2) = 0
answers: x = , x = 2
Solve 4x2 - 4x + 1 = by completing the square
4(x2 - x ) = (lots of steps!)
(x - )2 =
x = +
x = 2 and x = -1
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Quadratic Equations by Dr. Terri

14. Solving Quadratics: Exercises page 2
Solve 7x2 + 5 = by the quadratic formula
x = -b   b2 - 4ac 2a
x = -0   a = 7, b = 0, c = 5 14
x =   -140
x =  2i  35 = + i  35 . .
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Quadratic Equations by Dr. Terri