Quadratics 3102.3.30 Solve quadratic equations using multiple methods: factoring, graphing,...
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Transcript of Quadratics 3102.3.30 Solve quadratic equations using multiple methods: factoring, graphing,...
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!!
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!
Finding the solutions after graphing!
To find the solutions of the quadratic equation on a graph, look where the parabola intersects the x-axis!
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
The Quadratic Formula!!!
Yet another way to solve a quadratic equation is to use 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!
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