Download Quadratic Equations, Difference of Two Squares and Perfect Square

Algebra I Part 2
Unit 3 Day 6 - Quadratic Equations, Difference of Two Squares and Perfect Square
Trinomials (8-5, 8-6)
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Objective:
Factor binomials that are the difference of squares. Use the difference of squares to solve equations. Factor
perfect square trinomials. Solve equations involving perfect squares.
Factor Difference of Squares:
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Recall that in the last unit we learned about the product of the sum and difference of two quantities.
This resulting product is referred to as the difference of two squares.
So, the factored form of the difference of squares is called the product of the sum and difference of
the two quantities.
a2 – b2 = ____________________
x2 – 25 = ____________________
t2 – 64 = ____________________
Example 1: Factor each polynomial.
a.
m2 – 64
b. 16y2 – 81z2
c. 3b3 – 27b
Check Your Progress: Choose the best answer for the following.
A. Factor the binomial b2 – 9.
A. (b + 3)(b + 3)
B. (b – 3)(b + 1)
C. (b + 3)(b – 3)
D. (b – 3)(b – 3)
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B. Factor the binomial 25a2 – 36b2.
A. (5a + 6b)(5a – 6b)
B. (5a + 6b)2
C. (5a – 6b)2
D. 25(a2 – 36b2)
C. Factor 5x3 – 20x.
A. 5x(x2 – 4)
B. (5x2 + 10x)(x – 2)
C. (x + 2)(5x2 – 10x)
D. 5x(x + 2)(x – 2)
To factor a polynomial completely, a technique may need to be applied more than once.
This also applies to the difference of squares pattern.
Example 2: Factor each polynomial.
a.
y4 – 625
b. 256 – n4
Check Your Progress: Choose the best answer for the following.
A. Factor y4 – 16.
A. (y2 + 4)(y2 – 4)
B. (y + 2)(y + 2)(y + 2)(y – 2)
C. (y + 2)(y + 2)(y + 2)(y + 2)
D. (y2 + 4)(y + 2)(y – 2)
B. Factor 81 – d4.
A. (9 + d)(9 – d)
B. (3 + d)(3 – d)(3 + d)(3 – d)
C. (9 + d2)(9 – d2)
D. (9 + d2)(3 + d)(3 – d)
Algebra I Part 2
Unit 3 Day 6 - Quadratic Equations, Difference of Two Squares and Perfect Square
Trinomials (8-5, 8-6)
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Sometimes more than one factoring technique needs to be applied to ensure that a polynomial is
factored completely.
Example 3: Factor each polynomial.
a.
9x5 – 36x
b. 6x3 + 30x2 – 24x – 120
Check Your Progress: Choose the best answer for the following.
A. Factor 3x5 – 12x.
A. 3x(x2 + 3)(x2 – 4)
B. 3x(x2 + 2)(x2 – 2)
C. 3x(x2 + 2)(x + 2)(x – 2)
D. 3x(x4 – 4x)
B. Factor 5x3 + 25x2 – 45x – 225.
A. 5(x2 – 9)(x + 5)
B. (5x + 15)(x – 3)(x + 5)
C. 5(x + 3)(x – 3)(x + 5)
D. (5x + 25)(x + 3)(x – 3)
Solve Equations by Factoring:
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After factoring, you can apply the Zero Product Property to an equation that is written as the product
of factors set equal to 0.
Example 4: In the equation y = q2 -
4
, what is the value(s) of q when y = 0?
25
Check Your Progress: Choose the best answer for the following.
In the equation m2 – 81 = y, which is a value of m when y = 0?
A. 0
1
B.
9
C. -9
D. 81
Factor Perfect Square Trinomials:
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In a previous unit, we learned the patterns for the products of the binomials (a + b)2 and (a – b)2.
Recall that these are special products that follow specific patterns.
(a + b)2 = (a + b)(a + b)
= ____________________
= ____________________
(a – b)2 = (a – b)(a – b)
= ____________________
= ____________________
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Algebra I Part 2
Unit 3 Day 6 - Quadratic Equations, Difference of Two Squares and Perfect Square
Trinomials (8-5, 8-6)
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These products are called perfect square trinomials, because they are the squares of binomials.
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For a trinomial to be factorable as a perfect square, the first and last terms must be ____________
squares and the middle term must be two times the square roots of the first and last terms.
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
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Example 5: Determine whether each trinomial is a perfect square trinomial. Write yes or no. If it is a
perfect square, factor it.
a.
25x2 – 30x + 9
b. 49y2 + 42y + 36
Check Your Progress: Choose the best answer for the following.
A. Determine whether 9x2 – 12x + 16 is a perfect square trinomial. If so, factor it.
A. Yes; (3x – 4)2
B. Yes; (3x + 4)2
C. Yes; (3x + 4)(3x – 4)
D. Not a perfect square trinomial.
B. Determine whether 49x2 + 28x + 4 is a perfect square trinomial. If so, factor it.
A. Yes; (4x – 2)2
B. Yes; (7x + 2)2
C. Yes; (4x + 2)(4x – 4)
D. Not a perfect square trinomial.
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A polynomial is completely factored when it is written as a product of __________ polynomials.
More than one method might be needed to factor a polynomial completely.
Remember, if the polynomial does not fit any pattern or cannot be factored, the polynomial is
__________.
o Step 1: Factor out the GCF.
o Step 2: Check for a difference of squares or a perfect square trinomial.
o Step 3: Apply the factoring patterns for x2 + bx + c or ax2 + bx + c (general trinomials), or factor by
grouping.
Example 6: Factor each polynomial if possible. If the polynomial cannot be factored, write prime.
a.
6x2 – 96
b. 16y2 + 8y – 15
Algebra I Part 2
Unit 3 Day 6 - Quadratic Equations, Difference of Two Squares and Perfect Square
Trinomials (8-5, 8-6)
Check Your Progress: Choose the best answer for the following.
A. Factor the polynomial 3x2 – 3.
A. 3(x + 1)(x – 1)
B. (3x + 3)(x – 1)
C. 3(x2 – 1)
D. (x + 1)(3x – 3)
B. Factor the polynomial 4x2 + 10x + 6.
A. (3x + 2)(4x + 6)
B. (2x + 2)(2x + 3)
C. 2(x + 1)(2x + 3)
D. 2(2x2 + 5x + 6)
Solve Equations with Perfect Squares:
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When solving equations involving repeated factors, it is only necessary to set one of the repeated
factors equal to zero.
Example 7: Solve 4x2 + 36x = -81.
Check Your Progress: Choose the best answer for the following.
Solve 9x2 – 30x + 25 = 0
A.
5 
 
3 
 5
 
 3
C. {0}
D. {-5}
B.
Square Root Property:
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You have solved equations like x2 – 16 = 0 by factoring.
You can also use the definition of a square root to solve the equation.
x2 – 16 = 0
Original Equation
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Add 16 to each side.
__________
Take the square root of each side.
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Remember that there are two square roots of 16, namely ___ and ___.
Therefore, the solution is is {___, ___}.
You can express this as {_____}.
In the equation x2 = n, if n is not a perfect square, you need to approximate the square root.
Use a calculator to find an approximation.
If n is a perfect square, you will have an exact answer.
Example 8: Solve each equation. Check the solutions.
a.
(b – 7)2 = 36
b. (x + 9)2 = 8
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Algebra I Part 2
Unit 3 Day 6 - Quadratic Equations, Difference of Two Squares and Perfect Square
Trinomials (8-5, 8-6)
5
Check Your Progress: Choose the best answer for the following.
A. Solve the equation (x – 4)2 = 25. Check your solution.
A. {-1, 9}
B. {-1}
C. {9}
D. {0, 9}
B. Solve the equation (x – 5)2 = 15. Check your solution.
A.
B.
5  15
5  15
C. {20}
D. {10}
Example 9: A book falls from a shelf that is 5 feet above the floor. A model for the height h in feet if an
object dropped from an initial height of h0 feet is h = -16t2 + h0, where t is the time in seconds
after the object is dropped. Use this model to determine approximately how long it took for the
book to reach the ground.
Check Your Progress: Choose the best answer for the following.
An egg falls from a window that is 10 feet above the ground. A model for the height h in feet of an object
dropped from an initial height of h0 feet is h = -16t2 + h0, where t is the time in seconds after the object is
dropped. Use this model to determine approximately how long it took for the egg to reach the ground.
A. 0.625 second
B. 10 seconds
C. 0.79 second
D. 16 seconds