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 = 0Add 18 to both sides 2x² = 18Divide both sides by 2x² = 9Find the square root of both sides

x = ± 3

Example 2

Solve: 2x² + 72 = 0Subtract 72 from both sides 2x² = -72Divide both sides by 2 x² = -36Find 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 = 17Find the radius of a circle whose area is 125 in2.81x2 - 49 = 03x2 – 85 = 2x2 – 364x2 + 72 = 2x2 - 28

The Quadratic Formula

xb b ac

a

2 4

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 = 0a = 2; b = 4; c = 3b2 – 4ac(4)2 – (4)(2)(3)16 – 24= -8The answer is negative, so there are no real solutions.

Find the number of solutions:

2x2 – 11x + 6 = 0a = 2; b = -11; c = 6b2 – 4ac(-11)2 – (4)(2)(6)121 – 48= 73The answer is positive, so there are two real solutions.

Find the number of solutions:

2x2 + 12x = -182x2 + 12x + 18 = 0a = 2; b = 12; c = 18b2 – 4ac(12)2 – (4)(2)(18)144 – 1444= 0The answer is zero, so there is one real solution.

Try these…

3x + 2x2 + 6 = 0

3x2 = 13x – 4

9x2 + 16 = -24x

Quadratic Formula

xb b ac

a

2 4

2

6x2 + 7x – 5 = 0

a = 6, b = 7, c = -5

x 7 49 120

12

x 7 169

12

x 7 13

12

x x

7 13

12

7 13

12,

x x 6

12

20

12,

x x 1

2

5

3,

x 7 7 4 6 5

2 6

2( ) ( )( )

( )

5x2 – 4x = 33

-33 -335x2 – 4x – 33 = 0a = 5, b = -4, c = -33

x ( ) ( ) ( )( )

( )

4 4 4 5 33

2 5

2

x 4 16 660

10

x4 676

10

x4 26

10

x x

4 26

10

4 26

10,

x x 30

10

22

10,

x x

311

5,

6x2 – 150 = 0

Since b = 0, I would use the square root method.Add 130 to both sides 6x2 = 150Divide both sides by 6 x2 = 25Find 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