UNIT 2- Quadratic Relations & Equations
MA 40: Algebra 2
Task #3 – Factor & Solve Quadratic Equations
Common Core: HS.F.IF.8.a
Name:_______________
Date:_______Period:___
Factoring Quadratics
Factoring is a powerful tool used to solve many quadratic equations.
Factoring changes the quadratic from addition or subtraction form to
multiplication form. When the product is equal to zero, the solution is
based on the zero product property. Factoring is an algebraic method for
solving.
Zero Product Property: If a and b are any numbers and
(a)(b) = 0, then a or b or both must be equal to zero.
Example: Solve the equation x(x+5) = 0
The two factors of this equation are x and x+5. The zero product
property states one of these factors must equal zero.
That is, x = 0 or x + 5 = 0
Solving each equation gives the solutions: x = 0 or x = -5
The solutions of an equation occur where the graph touches or crosses
the x-axis. Therefore, solutions of the equation are also called zeros of
the function. The zeros on the graph are the horizontal intercepts. Use
your graphing calculator to graph Y1 = x(x + 5) and verify the zeros
graphically that were found algebraically. (Sketch a picture of your graph
and label the zeros)
Any factoring technique must begin with these steps:
Set the equation equal to zero.
Factor out the GCF (greatest common factor)
1. Set the equations equal to zero and factor out the GCF
a. x2 – 6x = 0
b. 6x2 + 14x = 30
c. 2x3 – 8x2 + 20x = -8x
You may recall that you have had previous experience with factoring out
the GCF and factoring quadratic equations if a = 1. The following activity
will lead you through factoring when a ≠ 1
2. Factor: 6𝑥 2 − 7𝑥 − 3
a. Think of the polynomial as fitting the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
What is a? ______
What is c? ______
What is the product ac? ______
b. List all possible pairs of integers such that their product is equal to
the number ac. Organize the list in a table. Make sure that your
integers are chosen so that their product has the same sign, positive
or negative, as the number ac from above, and make sure that you
list all the possibilities.
Factors of ac (-18)
Sum of factors
c. What is b in the quadratic polynomial given? ______
Add the integers from each pair listed in part b. Which pair adds to
the value of b from your quadratic polynomial? We will refer to these
integers as m and n.
m = _____ n = _____
d. Rewrite the polynomial replacing bx with mx + nx [Note either m
or n could be negative. The expression indicates to add the terms mx
and nx including the correct sign.]
e. Factor the polynomial from part 4 by grouping.
f. Check your answer by performing the indicated multiplication in
your factored polynomial. Did you get the original polynomial back?
3. Use the method outlined in the previous steps to factor each of
the following quadratic polynomials. Is it always necessary to list all
the integer pairs whose product is ac? Explain your answer.
a.
6𝑥 2 + 7𝑥 − 20
b.
2𝑥 2 + 3𝑥 − 54
c.
4𝑤 2 − 11𝑤 + 6
d.
3𝑡 2 − 13𝑡 − 10
e. 8𝑥 2 + 5𝑥 − 3
f.
18𝑥 2 + 17𝑥 + 4
g. 6𝑝2 − 49𝑝 + 8
4. Compare your factorization with other groups. Did everyone write
their answers the same way? Explain why answers can look different and
yet be equivalent.
5. If a quadratic polynomial can be factored in the form (Ax + B)(Cx + D)
where A, B, C, and D are all integers, the method you have been using will
lead to the answer, specifically called the correct factorization. Show that
each of these cannot be factored into the form (Ax + B)(Cx + D) using
integer coefficients.
a. 4z2 + z – 6 = 0
b. t2 + 2t = -8
c. 3x2 – 12 = -15x
6. As mentioned in the opening sentence of this activity, the purpose of
factoring is to solve the quadratic equation. Use your factorization from
Problem 3 and the zero product property to solve these quadratic
equations
a. 2x2 + 3x -54 = 0
b. 4w2 + 6 = 11w
c. 3t2 - 13t = 10
d. 2x(4x + 3) = 3 + x
e. 18z2 + 21z = 4(z – 1)
f. 8 – 13p = 6p(6 – p)
g. 24q2 = 4q + 8
Summary :
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