Quadratics

    3102.3.30  Solve quadratic equations using multiple methods: factoring,

graphing, quadratic formula, or square root principle.

What is a Quadratic??

The standard form for any quadratic equation is ax2 + bx + c = 0

There are many ways to solve quadratic equations. Below are the ways that we will solve them.

1. Factoring2. Graphing3. Quadratic Formula4. Square root principle

First, FACTORING!!

We will first factor a quadratic expression and solve for the unknown in the equation.

We will begin to factor quadratic expressions with a = 1.

First problem!

x2 + 7x + 12 = 0

( x ) ( x ) =0Now set each set of parentheses equal to

zero.

x + 3 = 0 x + 4 = 0 x = -3 x = -4

Put your answers in set notation

{ -3 , -4}

3 4

means same signs!Always goes with the largest number!!

++

Now let’s try this one!

x2 + 8x - 20 = 0

( )( ) =0

x x10 2

Means different signsAlways goes with the largest number

+ -

Second problem with a ≠ 0.

5x2 + 27x +10 = 05x2 + 25x + 2x +10 =0(5x2+25x)+(2x+10)=05x(x + 5) + 2(x + 5) =0 (x + 5) (5x + 2) = 0x + 5 = 0 5x + 2 = 0x = -5 x = -2/5

{-5,-2/5}

First, multiply a and c together. 5 * 10 = 50

Second, Ask yourself what are the factors of 50 that will add or subtract to give you b? –

Let’s list them1*50 = 50 2*25 = 505*10 = 50Which set of factors can add to give you

27? Correct! 2 and 25

Therefore we will have +25 and +2Now, group so it will be easy to factor!Since there are two (x+5), write them

one time! And also write the GCF of each in a set of parenthesis by themselves!

LAST, set each factor equal to zero and solve!

SAME SIGNS!!!!!

ALWAYS GOES WITH THE LARGER NUMBER!!

Try this one!

24x2 – 22x + 3 = 0

ANSWER: (4x – 3)(6x – 1) = 0

{3/4,1/6}

GRAPHING

Another way to solve a quadratic equation is to graph it! To graph a quadratic equation, you must have a domain. You can pick your own domain or it will be given!

y = 2x2 – 4x – 5

X Y

-2

-1

0

1

2

3

4

11

1

-5

-7

-5

1

11

vertex

Finding the solutions after graphing!

To find the solutions of the quadratic equation on a graph, look where the parabola intersects the x-axis!

Here is an example using the calculator!

y = x2 – x – 2

So, the solutions are -1 and 2

{-1, 2}

Or just look at the table!!

The solutions or ROOTS can be found in the table by finding where the y value is zero!

And the ROOTS are {-1, 2}

Looking at graphs!

You can look at graphs and tell how many roots they have! Here’s how!!

This graph has no solution because the parabola NEVER crosses the x-axis

This graph touches the x-axis one time. Therefore, we say that it has a double root!!

The Quadratic Formula!!!

Yet another way to solve a quadratic equation is to use the QUADRATIC FORMULA!!!!!!

The Quadratic Formula

Using the Quadratic Formula

X2 – 2x – 24 =0

A= 1 B= -2 C= -24

Now plug in the numbers into the formula!

2( 2) ( 2) 4(1)( 24)

2(1)x

Now just plug this into the calculator!

Calculator stepsNotice the addition sign

Notice the subtraction sign

The Square Root Principle

Another way to solve a quadratic equation is to use the square root principle!

Some equations can be solved by taking the square root of both sides!

Using the square root principle!

x2 – 10x + 25 = 7

( )( )=7

(x – 5)2 = 7

Steps:

1. Factor the left side.

2. Write the factors one time

3. Now take the square root of both sides.

4. Now solve the two problems! (Use your calculator)

x x5 5- -

2( 5) 7x

5 7x 5 7x 5 7x

Calculator steps

5 7x 5 7x

x= 7.6 x= 2.4

{2.4,7.6}

Pick 3 ways out of the 4 to solve this problem!x2 – 5x – 24 = 0

The answer you should get if you work the problem three ways is

{-3,8}