Section 6.7
A Solving a Quadratic Equation
by Factoring
In this section we will use the methods of
factoring, and a special property of 0, to solve
quadratic equations.
The number 0 has a special property. If we
multiply two numbers and the product is 0, then
one or both of the original two numbers must be 0.
To solve the quadratic equation :
x2 + 5x + 6 = 0
Factor the left side into: (x + 2)(x + 3) = 0
Now, (x + 2) and (x + 3) both represent real
numbers and their product is 0
Therefore, either (x + 3) is 0 or (x + 2) is 0.
We use the Zero Factor Property to finish the
problem:
x2
+ 5x + 6 = 0
(x + 2)(x + 3) = 0
x+2=0
or
x+3=0
x = –2 or
x = –3
Factor the trinomial
Apply Zero Factor Property
Solve resulting equations
The equation has two solutions.
The solution set is {–2, –3}.
To check our solutions we have to check each one
separately to see that each one produces a true
statement when used in place of the variable:
When
x = –3
the equation
x2 + 5x + 6 = 0
becomes
(–3)2 + 5(–3) + 6 ≟ 0
9 + (–15) + 6 ≟ 0
0 =0
When
the equation
becomes
x = –2
x2 + 5x + 6 = 0
(–2)2 + 5(–2) + 6 ≟ 0
4 + (–10) + 6 ≟ 0
0=0
Step 1: Put the equation in Standard Form, that is,
isolate zero on one side and arrange
decreasing powers of the variable on the other.
Step 2: Factor completely.
Step 3: Apply the Zero Factor Property by setting each
factor equal to zero.
Step 4: Solve the resulting equations.
Step5: You can check both solutions by plugging each
into the original equation.
Solve the equation: 2x2 – 5x = 12.
2x2 – 5x = 12
2x2 – 5x – 12 = 0
(2x + 3)(x – 4) = 0
2x + 3 = 0 or x – 4 = 0
2x = –3
x=4
x=
Put equation in standard form
by adding –12 to both sides
Factor left side completely
Apply zero factor property
Solve resulting equations
Substitute each solution into 2x2 – 5x = 12 to check:
2x2
2
2
– 5x
= 12
2x2 – 5x = 12
–5
≟ 12
2(4)2 – 5(4) ≟ 12
+5
= 12
2(16) – 20 = 12
+
= 12
32 – 20 = 12
= 12
12 = 12
12 = 12
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Section 6.7
Pages 465-468
# 1, 11, 13, 15, 21, 27, 37, 55, 65