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Thursday 19th of September
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 Depdendent Variable

 Number of equations to solve: 23456789
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 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

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 Solve for:

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I. Finding Roots of Quadratic Equations
a. The Standard Form of a quadratic equation is: ax2 + bx + c = 0 .
b. We can use the Quadratic Formula to solve equations in standard
form:

c. Discriminant – The radical portion of this formula sqrt(b2 − 4ac) ,
determines the nature of the roots. This quantity under the radical
sign b2 − 4ac , is called the discriminant.

d. Three things may occur regarding the discriminant:

i. If b2 − 4ac > 0
We can take the square root of this positive amount
and there will be two different real answers (or roots)
to the equation.

ii. If b2 − 4ac < 0
We cannot take the square root of a negative number,
so there will be no real roots.

iii. If b2 − 4ac = 0
The amount under the radical is zero and since the
square root of zero is zero, we will get only 1 distinct
real root.

II. Examples

III. Practice Problems
By examining the discriminant = b2 − 4ac , determine how many real
roots, if any, the following quadratic equations have.

1. x2 − 4x + 4 = 0

2. x2 + 4 = 0

3. x2 − 2x + 4 = 0

4. x2 − 4x = 0

5. 5r2 − 3r + 2 = 0

6. 7x2 −10x − 5 = 0

7. x2 − 4 = 0

8. 25t2 −10t = −1

9. 6y2 − 5y = 21

10. 2y2 −19y = 3