Download Solving Quadratic Equations

Solving Quadratic Equations
Cont’d.
To Solve A Quadratic Equation
When b = 0…
Use the same procedures you used to solve
an equation to get the “x” isolated (by itself).
Instead of having an “x” left, you have an
“x²”.
When the “x²” is isolated, find the square
root of both sides (be sure to give both the
principal and the negative roots!).
Example 1
Solve: 2x² - 18 = 0
Add 18 to both sides 2x² = 18
Divide both sides by 2 x² = 9
Find the square root of both sides
x=±3
Example 2
Solve: 2x² + 72 = 0
Subtract 72 from both sides 2x² = -72
Divide both sides by 2
x² = -36
Find the square root of both sides—
oops!! You can’t find the square root of
a negative number (-36) so there is
NO SOLUTION!
Try these…
4x2 + 1 = 17
Find the radius of a circle whose area is
125 in2.
81x2 - 49 = 0
3x2 – 85 = 2x2 – 36
4x2 + 72 = 2x2 - 28
The Quadratic Formula
 b  b  4ac
x
2a
2
The Discriminant
This is the part of the equation under
the radical sign. (b2 – 4ac)
When that is positive, there will be two
answers.
When that is negative, there will be no
real solution.
When that is zero, there will be one
answer.
Find the number of solutions:
2x2 + 4x + 3 = 0
a = 2; b = 4; c = 3
b2 – 4ac
(4)2 – (4)(2)(3)
16 – 24= -8
The answer is negative, so there are no
real solutions.
Find the number of solutions:
2x2 – 11x + 6 = 0
a = 2; b = -11; c = 6
b2 – 4ac
(-11)2 – (4)(2)(6)
121 – 48= 73
The answer is positive, so there are two
real solutions.
Find the number of solutions:
2x2 + 12x = -18
2x2 + 12x + 18 = 0
a = 2; b = 12; c = 18
b2 – 4ac
(12)2 – (4)(2)(18)
144 – 1444= 0
The answer is zero, so there is one real
solution.
Try these…
3x + 2x2 + 6 = 0
3x2 = 13x – 4
9x2 + 16 = -24x
Quadratic Formula
 b  b  4ac
x
2a
2
6x2 + 7x – 5 = 0
a = 6, b = 7, c = -5
 7  (7) 2  4(6)( 5)
x
2(6)
 7  49  120
x
12
x
 7  169
12
 7  13
x
12
 7  13
 7  13
x
, x
12
12
x
6
 20
, x
12
12
1
5
x , x
2
3
5x2 – 4x = 33
-33
-33
5x2 – 4x – 33 = 0
a = 5, b = -4, c = -33
 ( 4)  ( 4) 2  4(5)( 33)
x
2(5)
4  16  660
x
10
4  676
x
10
x
4  26
10
4  26
4  26
x
, x
10
10
30
 22
x , x
10
10
 11
x  3, x 
5
6x2 – 150 = 0
Since b = 0, I would use the square
root method.
Add 130 to both sides
6x2 = 150
Divide both sides by 6
x2 = 25
Find the square root of both sides and
round to the nearest hundredth.
x = 5, and –5
Try these…
x2 + x – 12 = 0
2x2 + x – 7 = 0