M 1310
2.3
Quadratic Equations
1
Quadratic Equations have the form ax 2 + bx + c = 0 ,
where a ≠ 0 .
Methods for solving quadratic equations:
•
•
•
•
factoring
taking square roots
completing the square
using the quadratic formula
Solve by factoring:
1. Move everything to one side of the equation and be
sure to make the x 2 term is positive.
It will have the form ax 2 + bx + c = 0
2. Factor the left-hand side of the equation.
3. Set each factor equal to zero.
4. Solve each of the resulting equations.
Example 1:
Solve the equation.
x 2 − 3x = 4
M 1310
2.3
Quadratic Equations
Example 2:
Solve the equation.
− 3 x 2 = −3 x − 60
Example 3:
Solve the equation.
3x 2 + 5x + 2 = 0
Example 4:
Solve the equation.
3x 2 − 10 x − 8 = 0
2
M 1310
2.3
Quadratic Equations
Square Root Method
Equation has the form of ax 2 − b = 0 , where a and b
have to be perfect squares at least in this course.
Example 5:
Solve the equation.
x 2 − 25 = 0
Example 6:
Solve the equation.
16 x 2 − 25 = 0
3
M 1310
2.3
Quadratic Equations
4
Solving by Completing the Square
1. Write the equation in the form x 2 + bx = c .
2
⎛b⎞
2. Add ⎜ ⎟ to both sides.
⎝2⎠
2
⎛b⎞
⎛b⎞
x + bx + ⎜ ⎟ = c + ⎜ ⎟
⎝2⎠
⎝2⎠
3. Factor the left-hand side.
4. Solve by the square root method.
2
2
Example 7:
Solve the equation.
x 2 − 6 x = 13
Example 8:
Solve the equation.
x 2 + 4 x − 21 = 0
M 1310
2.3
Quadratic Equations
Example 9:
Solve the equation.
x 2 − 12 x = 0
Solving by the Quadratic Formula
− b ± b 2 − 4 ac
x=
2a
Example 10:
Solve the equation.
x 2 − 6x + 8 = 0
5
M 1310
2.3
Quadratic Equations
6
Example 11:
Solve the equation.
3 x 2 + 2 x = −2
Note:
The discriminate is the part of the quadratic formula
under the square root.
b 2 − 4 ac
Depending on the value of this part of the equation it
can tells us what the solutions to the equation look like.
If b 2 − 4 ac < 0
(next section)
no real solutions
If b 2 − 4 ac > 0
2 real solutions
If b 2 − 4 ac = 0
1 real solution