Transcript Quadratics
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. Factoring
2. Graphing
3. Quadratic Formula
4. 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!
Always goes with the
largest number!!
x2 +
means same signs!
7x +
12
=0
( x + 3 ) ( x + 4 ) =0
Now set each set of parentheses equal to
zero.
x+3=0
x = -3
x+4=0
x = -4
Put your answers in set notation
{ -3 , -4}
Now let’s try this one!
Always goes with the
largest number
x2
Means different signs
+
8x
( x + 10 )( x
-
20
=0
-
2 )
=0
Second problem with a ≠ 0.
ALWAYS GOES
WITH THE
LARGER
NUMBER!!
5x2 +
SAME SIGNS!!!!!
27x +10 = 0
5x2 + 25x + 2x +10 =0
(5x2+25x)+(2x+10)=0
5x(x + 5) + 2(x + 5) =0
(x + 5) (5x + 2) = 0
x+5=0
5x + 2 = 0
x = -5
x = -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 them
1*50 = 50
2*25 = 50
5*10 = 50
Which set of factors can add to give you
27?
Correct! 2 and 25
Therefore we will have +25 and +2
Now, group so it will be easy to factor!
{-5,-2/5}
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!
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
11
-1
1
0
-5
1
-7
2
-5
3
1
4
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) 4(1)(24)
x
2(1)
2
Now just plug this into the
calculator!
Calculator steps
Notice 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
(x - 5 )(x - 5)=7
(x – 5)2 = 7
( x 5) 7
2
x 5 7
x 5 7 x 5 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)
Calculator steps
x 5 7
x 5 7
x= 2.4
x= 7.6
{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}