Math 2 +2C Handout
Solving Quadratic Equations With Real Solutions
Name____________________________
Instructor_________________________
Date_____________________________
Learning Objectives:
• Objective 1: General form of a quadratic equation
• Objective 2: Methods of solving quadratic equations
Objective 1: General form of a quadratic equation
The general form of a quadratic equation is ax 2 + bx + c =
0, a ≠ 0
1. Determine whether the equation is quadratic or not? If yes, identify a, b and c.
a. 2 x 2 − x + 1 =0
b. 3 y 2 = 1
c. −t 2 =
2t
1/2
d. 2 x + 2 x + 3 =
0
Objective 2: Methods of solving quadratic equations
We can solve a quadratic equation using four different methods.
Method 1: Solving quadratic equations by factoring
Zero Factor Property: If ab = 0 , then
=
a 0=
or b 0 .
2. Solve the following equations by factoring.
a. (2 x − 1)(4 x + 5) =
0
2
b. 3U + 10U − 10 =
−2
2
c. 3 x − 15 x − x + 5 =
0
2
d. 10 z + 35 z =6 z + 21
e. x 3 = x
f. 9 x 2 + 4 =
12 x
3
2
g. 2 x + 8 x =
10 x
Method 2: Solving quadratic equations by using the Principle of Square Roots.
Principle of Square Roots: For any real number a, If x 2 = a , a ≥ 0 , then x = a or x = − a
3. Solve the following equation using Principle of Square Roots.
a. 2 x 2 − 3 =
0
2
b. 5 y + 1 =
8
2
c. 4 x + 1 =
0
2
d. ( P − 1) + 2 =
3
e. 3(2 x + 1) 2 + 2 =
7
Method 3: Solving quadratic equations by completing the square:
2
b
Rule: To complete the square for x + bx or x − bx , add   .
2
2
2
2
b
4. Complete the square by adding   and write the expression as
2
2
a. x + 12 x + ______ =
(
)2
b. x 2 − 7 x + ______ =
(
5
(
c. x 2 + x + ______ =
2
1
(
d. x 2 + x + ______ =
3
2
2
b
b


 x +  or  x −  .
2
2


)2
)2
)2
Example: Solve 2 x 2 + 8 x − 4 =
0 by completing the square.
Answer:
i) Divide all terms by the coefficient of x 2 and move the constant term to the other side.
x2 + 4x − 2 =
0
2
x + 4x =
2
2
b
ii) Add   to both sides of the equation.
2
x2 + 4x + 4 = 2 + 4
iii) Write one side as a perfect square and simplify the other side.
( x + 2) 2 =
6
iv) Use square root property to solve the equation.
x + 2 =± 6
5. Solve the following quadratic equations by completing the square.
a. x 2 + 5 x − 3 =
0
2
b. 3 x − 5 x − 2 =
0
2
c. 5 y + 4 y − 3 =
0
3
1
d. x 2 + x =
2
4
2
e. 2 x − 3 x − 1 =0
Method 4: Solving quadratic equations by using quadratic formula:
−b ± b 2 − 4ac
2a
6. Solve the following quadratic equations using quadratic formula.
a. 2 x 2 + 3 x =
5
2
b. y + 2 y =
4
c. 3u (u + 2) =
1
Quadratic Formula: If ax 2 + bx + c =
0 a ≠ 0 , then x =
d. 3 y 2 − 7 y =
−2
7. Solve the following quadratic equations using an appropriate method.
a. 2 x 2 − 7 x + 3 =
0
2
b. 2 x − x =
15
2
c. (2 x − 5) − 4 =
0
2
d. x − 2 x =
15