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( x + 4) ( x + 6 )
( x – 7) ( x – 3 )
( x – 2) ( x + 8 )
Algebra Unit – Part 1
( x + 1) ( x - 5 )
• Why did you learn to factor a trinomial?
You learned to factor a trinomial into TWO binomials in order to use those answers.
In order to use those answers you have to set each binomial = to zero (0).
( X – 3 ) ( x + 5 ) = 0
If you multiply two binomials and the value of one of them is zero, then the whole product is zero.
• Setting the binomial = to Zero
The product ( x – 3 ) ( x + 5 ) = 0
If either binomial = zero then the whole things is zero.
What number would make the 1st binomial = 0?
3
What number would make the 2nd binomial = 0?
-5
That means if x = 3 or x = -5, the whole problem is 0. Therefore, our answer is x = 3 or x = -5.
• Sample Problems
In the middle of your sheet of notes fill in the answers as we go, by setting each binomial = to zero.
( x – 2 ) ( x + 8) = 0
X = 2 or x = -8
( x – 7 ) ( x – 3 ) = 0
X = 7 or x = 3
( x + 4 ) ( x + 6 ) = 0
X = -4 or x = -6
( x + 1 ) ( x – 5 ) = 0
X = -1 or x = 5
• Solving Basic Square Root Problems
The easiest type of quadratics to solve is basic square root problems.
They come in two forms:
x2= 64 and x2 – 36 = 0
To solve the first one, all you do is take the square root of the number. X = √64 = 8 and -8
To solve the second one, you have to add 36 to both sides and then take the square root.
x2 – 36 = 0
+36 +36
x2 = 36
X = √36 = 6 and-6
• Sample Problems
At the end of your sheet of notes fill in the answers as we go, by setting each binomial = to zero.
• x2= 81
• X = √81 = 9 and-9
• x2= 9
• X = √9= 3 and-3
• x2 = 16
• X = √16 = 4 and-4
x2 – 25 = 0
+25 +25
x2 = 25
X = √25 = 5 and-5
x2 – 49 = 0
+49 +49
x2 = 49
X = √49 = 7 and-7
x2 – 4= 0
+4 +4
x2 = 4
X = √4= 2 and-2
• Steps to solving regular quadratics
Set the trinomial = to zero.
Factor the trinomial into the product of binomials
Set each binomial = to zero
Solve for x
Example
Solve: x2+ 5x -24 = 0
Only way to get a negative at the end is multiply 1 positive & 1 negative, looking at middle number the bigger number needs to be positive.
( x + 8 ) ( x - 3) = 0
( x + 8 ) = 0 or ( x – 3) = 0
X = -8 or x = 3
In the middle of your sheet of notes fill in the answers as we go, by creating binomials and then setting each binomial = to zero.
x2+ 12x + 32= 0
( x + 8 ) ( x + 4) = 0
( x + 8 ) = 0 or ( x + 4) = 0
X = -8 or x = -4
X2- 8x + 15= 0
( x - 5 ) ( x - 3) = 0
( x - 5 ) = 0 or ( x - 3) = 0
X = 5or x = 3
• X2- 8x - 9 = 0
• ( x - 9 ) ( x + 1) = 0
• ( x - 9 ) = 0 or ( x + 1) = 0
• X = 9 or x = -1
Again the first thing you need to do is set the trinomial = to zero. Therefore you may need to add or subtract in order to do so.
Example
x2+ 13x = -30
+30 + 30
x2+ 13x + 30= 0
( x + 10 ) ( x + 3) = 0
( x + 10 ) = 0 or ( x + 3) = 0
X = -10 or x = -3
• Sample one more step problems
At the end of your sheet of notes fill in the answers as we go, by setting the trinomials equal to zero, creating binomials and then setting each binomial = to zero.
Example 2
x2 + 2x = 63
-63 -63
x2+ 2x - 63= 0
( x + 9) ( x - 7) = 0
( x + 9 ) = 0 or ( x - 7) = 0
X = -9 or x = 7
Example 1
x2 + 8x = -12
+12 + 12
x2+ 8x + 12= 0
( x + 6) ( x + 2) = 0
( x + 6) = 0 or ( x + 2) = 0
X = -6or x = -2
• Example 3
• x2 - 14x + 60 = 12
• -12 -12
• X2 -14x +48 = 0
• ( x - 8) ( x - 6) = 0
• ( x - 8 ) = 0 or ( x - 6) = 0
• X = 8 or x = 6