# Roots of Quadratic Equations

**I. Finding Roots of Quadratic Equations**

a. The Standard Form of a quadratic equation is: ax^{2} + 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(b^{2} − 4ac) ,

determines the nature of the roots. This quantity under the radical

sign b^{2} − 4ac , is called the discriminant.

d. Three things may occur regarding the discriminant:

i. If b^{2} − 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 b^{2} − 4ac < 0

We cannot take the square root of a negative number,

so there will be no real roots.

iii. If b^{2} − 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 = b^{2} − 4ac , determine how many real

roots, if any, the following quadratic equations have.

1. x^{2} − 4x + 4 = 0

2. x^{2} + 4 = 0

3. x^{2} − 2x + 4 = 0

4. x^{2} − 4x = 0

5. 5r^{2} − 3r + 2 = 0

6. 7x^{2} −10x − 5 = 0

7. x^{2} − 4 = 0

8. 25t^{2} −10t = −1

9. 6y^{2} − 5y = 21

10. 2y^{2} −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