solving quadratic equations. what is the definition of a solution to a quadratic equation? a...

Solving Quadratic Equations

2 5 4 0x x

2 100x 2 4

2

b b acx

a

2( 1) 25x

Solving Quadratic EquationsWhat is the definition of a solution to a quadratic equation?

2y ax bx c A solution is the value of x when y = 0.

What are other terms for the solutions to a quadratic equation?

02ax bx c

x-intercepts(in some cases)

One x-intercept = one real solution (always rational)

Two x-intercepts = two real solutions (rational or irrational)

No x-intercepts = two complex

solutions

Graphs and Solutions to Quadratic Functions: 3 Cases

ALL QUADRATIC EQUATIONS CAN

BE SOLVED!

Solving QuadraticsName the 4 methods of solving quadratics

( )( ) 0x x

Method 1: Solving Quadratic Equations by Factoring

Let's solve the equation 1872 xx1. First you need to get it in what we call “standard form" which means 02 cbxax

need this to be 01

ok

01872 xx2. Now let's factor the left hand side 029 xx

3. Now set each factor = 0 and solve for each answer.

02or 09 xx

2or 9 xx

02 cbxaxok

Meaning: 2 x-intercepts, 2 real solutions

Factoring is the easiest way to solve a

quadratic equations, but it won’t work for all

functions, as many cannot be factored!

Subtract 18

252 t

Method 2: The Square Root: ax2 + c = 0

This method will work for any equation that doesn’t have a “bx” term, it only has “ax2” or “a(x-h)2” and a constant. The objective is to get x2 alone on one side of the equation and then take the square root of each side to cancel out the square.

1255 2 t1. Get the "squared stuff" alone which in this case is the t 2

5 5

25

2. Now square root each side.

252 t

5t Don’t forget that (-5)(-5) = 25 also!

Meaning: 2 x-intercepts, 2 real solutions

2 49

4u

Let's try another one:

24 49 0u 1. Get the "squared stuff" alone which in this case is the u 2

4 42. Square root each side.

2 49

4u

Remember with a fraction you can square root the top and square root the bottom

DON'T FORGET BOTH THE + AND –2 49

4u

7

2

iu

Recall , so x equals two imaginary numbers!

Meaning: no x-intercepts, but there are still 2 solutions.

1i

24 49u

Hey, what about the – under the square root?

You try

23 36 0x

21 50x

Another Example: “a(x-h)2”

21 50 0x

1, Get the "squared stuff" alone (i.e, the parentheses)

2. Now square root each side and DON'T FORGET BOTH THE + AND –

Let's simplify the radical1 5 2x

25 · 2

Now solve for x

1 5 2x Meaning: two x-intercepts, but they are irrational.

-1 -1

Perfect Square Trinomials: What’s the pattern?

2( 1)x 2 2x x

2 4x x

2 6x x 2 8x x 2x bx

16

9

4

1

Factored formAdd how much? c = ?

2x bx

To complete the square and make a perfect square trinomial,_________________

2( 3)x

2( 2)x

2( 4)x 2

2

b

2( )2

bx

“add half of b squared”

What completes the square?

2 20 ___x x 100 2( 10)x

2 12 ___x x 36 2( 6)x

2 9 ___x x 81/429

( )2

x

23 24 ___x x No Solution

You can only complete the

square when a = 1!

20

/ 2 10

b

b

12

/ 2 6

b

b

9

/ 2 9 / 2

b

b

Method 4: The Quadratic Formula

The Quadratic Formula is a formula that can solve any quadratic, but it is best used for equations that cannot be factored or when completing the square requires the use of fractions. It is the most complicated method of the four methods.

Do you want to see where the formula comes from?

2

2

4

2 4

b b acx

a a

aaa

The Quadratic Formula2 0ax bx c

2ax bx c

2 b cx x

a a

2

24

b

a

2

24

b

a

2

2

bx

a

2 4

2

b ac

a

2 4

2 2

b b acx

a a

2 4

2

b b acx

a

This formula comes from completing the square of a quadratic written in standard form

1. Subtract c and Divide by a

2. Complete the square:

3. Factor left side, combine right side

4. Square root each side

5. Simplify radical

6. Get x alone

7. Simplify right hand side2 2

b b

a

2

2

4

4

b ac

a

“x equals opposite b plus or minus square root of b squared

minus 4ac all over 2a”

2 4

2

b b acx

a

2 4

2

b b acx

a

This part of the formula is called the “Discriminant”

2 4b ac

The discriminant tells us what kind of solutions we have:

0 One real solution one x -intercept (always rational)

Two real solutions two x-intercepts (rational or irrational)

two complex solutions

(no x-intercepts)

The Quadratic Formula24 2 5 0x x

2 4 80

8x

Solve the equation

1. Identify a, b, c

2. Plug into the formula

5. Simplify

6. Simplify

7. Simplify radical

2 4

2

b b acx

a

a = 4b= 2c = 5

(4)(2)22

(4)

(5)

2 76

8x

2 2 19

8

ix

Notice the solutions are complex!

8. Simplify final answer, if possible

2 2 19

8

ix

1 19

4

ix

=4•19

Meaning: 0 x-intercepts, 2 complex solutions

The Quadratic Formula2 1 0x x Solve the equation

1. Identify a, b, c

2. Plug into the formula

5. Simplify

6. Simplify

7. Simplify radical

2 4

2

b b acx

a

a =b= c =

( )( )2

( )

( )

8. Simplify final answer, if possible

The Quadratic Formula2 1 0x x Solve the equation

1. Identify a, b, c

2. Plug into the formula

5. Simplify

6. Simplify

7. Simplify radical

2 4

2

b b acx

a

a = 1b= 1c = -1

( 1 )( 1)21

( 1 )

(-1 )

8. Simplify final answer, if possible

1 1 4

2x

1 5

2x

Already simplified

Another example

Solve the equation 22 4 5 0x x

True or False?

All quadratic equations have solutions. “No solution” could never be an answer to a

quadratic equation.

TRUE. You can solve ANY quadratic equation, you just may need to use

a particular method to get to the answers.

True or False?

The solutions (zero, root) to a quadratic equation are always x-intercepts on its graph.

False All quadratics have solutions. It is the value of x when y = 0.

But not all quadratics cross the x – axis, so the solutions will not always be x-intercepts.

8

6

4

2

-2

-4

-6

-10 -5 5 10 15

g x = - x-4 2-1

f x = x2+4x+5

Neither of these functions have x-intercepts, but they still have two complex solutions

True or False?

All quadratics equations can be factored.

FalseHere are just a few examples of quadratics that cannot be factored:

2 4 0x 2 5 8 0x x

29 2 1 0x x

True or False?

False. You will need to have the formula memorized.

The quadratic formula will be provided to you for the test and final exam.

Which method should you use?Solve 2 6 5 0x x

a. (x+1)(x+5) b. x = -1, x = -5 c. x = 1, x = 5 d. no sol

Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?

( )( ) 0x x

Which method should you use?Solve 2 3 2 0x x

a. b.

c. no sol because you cannot factor it

3 17

2x

3 17

2x

Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?

( )( ) 0x x

Which method should you use?

Solve 22 1x

a. b.

c. no sol because you cannot factor it

1

2x 2

2x

Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?

( )( ) 0x x

Which method should you use?

Solve 24 2 1 0x x

a. b.

c. d.

2 8

8

ix

2 8

8x

2 2 2

8

ix

1 2

4

ix

Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?

( )( ) 0x x

Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?

Which method should you use?

29 81 0x a. b. c. 3x 9x 81x

( )( ) 0x x

Completing the Square: Use #1

This method is used for quadratics that do not factor, although it can be used to solve any kind of quadratic function.

1. Get the x2 and x term on one side and the constant term on the other side of the equation.

2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this.

3 14x

2 6 5 0x x

2 6 5x x

3. Factor the trinomial

9 92 6 9 14x x

2( 3) 14x

3 14x 4. Apply the square root and solve for x

Completing the Square: Use #2

By completing the square, we can take any equation in standard form and find its equation in vertex form: y = a(x-h)2 + k

1. Get the x2 and x term on one side and the constant term on the other side of the equation.

2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this.

( 3, 14)

2 6 5 0x x

2 6 5x x

3. Factor the trinomial.

9 92 6 9 14x x

2( 3) 14x 4. Write in standard form.

2( 3) 14y x

What is the vertex?