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Square Roots and Radical Expressions
Solving Radical Equations
Simplifying Radical Expressions
Irrational Numbers in General and Square Roots in Particular
Roots of Polynomials
Simplifying Radical Expressions
Exponents and Radicals
Products and Quotients Involving Radicals
Roots of Quadratic Equations
Radical Expressions
Radicals and Rational Exponents
Find Square Roots and Compare Real Numbers
Radicals
Radicals and Rational Exponents
Theorems on the Roots of Polynomial Equations
SYNTHETIC DIVISION AND BOUNDS ON ROOTS
Simplifying Radical Expressions
Exponents and Radicals
Properties of Exponents and Square Roots
Solving Radical Equations
Rational Exponents and Radicals,Rationalizing Denominators
Rational Exponents and Radicals,Rationalizing Denominators
Quadratic Roots
Exponents and Roots
Multiplying Radical Expressions
Exponents and Radicals
Solving Radical Equations
Solving Quadratic Equations by Factoring and Extracting Roots
Newton's Method for Finding Roots
Roots of Quadratic Equations Studio
Roots, Radicals, and Root Functions
Review division factoring and Root Finding
Radicals
Simplifying Radical Expressions
Multiplying and Simplifying Radical Expressions
LIKE RADICALS
Multiplication and Division of Radicals
Radical Equations
BOUNDING ROOTS OF POLYNOMIALS
   
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Roots of Quadratic Equations

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

Answers: Roots of Quadratic Equations
1. 1 real root
2. no real roots
3. no real roots
4. 2 real roots
5. no real roots
6. 2 real roots
7. 2 real roots
8. 1 real root
9. 2 real roots
10. 2 real roots

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