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Math 142 — Rodriguez
Lehmann — 7.3
Using the Square Root Property to Solve Quadratic Equations
So far we only know how to solve quadratic equations by factoring. In this section we will
learn another method for solving (certain) quadratic equations. Before we learn it though, we
have to discuss square roots and complex numbers.
I.
Square Roots
A square root of k is b if b2=k. Numbers have two square roots. We denote the principal (nonnegative) square root using _____ .
The square roots of 9 are _____ and ______.
If I use the radical symbol: 9 =
Note: We first saw roots in Chap 4 but we saw them written with rational exponents. That is
____ was written as _____.
Simplifying Square Roots
To simplify a radical expression we make sure:
• No radicand contains a perfect square.
• No radicand contains a fraction.
• No denominator contains a square root.
Perfect Squares are:
Product Property for Square Roots states: For a≥0 and b≥0,
ab = a ⋅ b .
Simplify.
12
63
48
90
50
Quotient Property for Square Roots states:
For a≥0 and b≥0,
a
a
.
=
b
b
Simplify.
7
36
3
18
7
12
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7
5
6
20
II. Complex Numbers
The imaginary unit, written as i, is the number whose square root is –1.
This means that i2=–1.
i = −1 .
A complex number is a number of the form a + bi where a and b are real numbers.
4 + 3i
20
7i
0.5 — 1.25i
Square Root of a Negative Number
For p a positive number:
−p = i p
−16
−24
−40
−
11
49
− −18
−
5
7
III. Solving Quadratic Equations using the Square Root Property
The only method we have so far to solve ax2 + bx + c = 0 is the Zero Factor Property
(poly=0; factor the poly; set each factor = 0). If the poly doesn’t factor we can’t use this
method so what do we do?
Recall: if bn = k, then
In particular, if b2 = k, then
Written using square roots instead of rational (fraction) exponents means:
This is known as the Square Root Property.
Lehmann − 7.3
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Steps:
1. Isolate the squared term on one side of the equation.
2. Apply the Square Root Property.
3. Solve for x. Simplify the answer as much as possible (but don’t approximate it).
Solve.
2
1. x − 32 = 0
2. 5x 2 − 3 = 0
3.
( x − 2 )2 = 18
4.
5.
( x − 3)2 = −45
6. −7 ( x + 8 ) + 4 = −1
7. 5w − 28 = 5
2
9.
( 2x + 1)2 = 16
2
8. 2 ( x − 4 ) + 5 = 27
2
(t + 8 )2 = − 48
Notice: When can we use the Square Root Property?
Lehmann − 7.3
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Applications
Lehmann − 7.3
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