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Taking the Square Root
of Both Sides
Adapted from Walch Education
Complex Numbers
• The imaginary unit i represents the non-real value
i = -1 . i is the number whose square is –1. We
define i so that i = -1 and i 2 = –1.
• A complex number is a number with a real
component and an imaginary component. Complex
numbers can be written in the form a + bi, where a
and b are real numbers, and i is the imaginary unit.
5.2.1: Taking the Square Root of Both Sides
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Real Numbers
• Real numbers are the set of all rational and irrational
numbers. Real numbers do not contain an imaginary
component.
• Real numbers are rational numbers when they can
be written as
m
, where both m and n are integers and
n
n ≠ 0. Rational numbers can also be written as a
decimal that ends or repeats.
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Irrational Numbers
• Real numbers are irrational when they cannot be
m
written as , where m and n are integers and n ≠ 0.
n
Irrational numbers cannot be written as a decimal
that ends or repeats. The real number
is an
irrational number because it cannot be written as the
ratio of two integers.
•
examples of irrational numbers include
5.2.1: Taking the Square Root of Both Sides
2 and π.
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Quadratic Equations
• A quadratic equation is an equation that can be
written in the form ax2 + bx + c = 0, where a ≠ 0.
• Quadratic equations can have no real solutions, one
real solution, or two real solutions. When a quadratic
has no real solutions, it has two complex solutions.
• Quadratic equations that contain only a squared term
and a constant can be solved by taking the square
root of both sides. These equations can be written in
the form x2 = c, where c is a constant.
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Quadratic Equations
c tells us the number and type of solutions for the
equation.
Number and type of
solutions
c
Negative
Two complex solutions
0
One real, rational
solution
Positive and a perfect
square
Positive and not a
perfect square
Two real, rational
solutions
Two real, irrational
solutions
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Practice # 1
Solve (x – 1)2 + 15 = –1 for x.
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Solution
Isolate the squared binomial.
(x – 1)2 + 15 = –1
(x – 1)2 = –16
Original equation
Subtract 15 from
both sides.
Use a square root to isolate the binomial.
x - 1= ± -16
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Solution, continued
Simplify the square root.
• There is a negative number under the radical,
so the answer will be a complex number.
x - 1= ± -16
x - 1= ± (-1)(16)
x - 1= ± -1• 16
x - 1= ±4i
Equation
Write –16 as a product of a
perfect square and –1.
Product Property of Square
Roots
Simplify.
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Solution, continued
Isolate x.
x - 1= ±4i
x = 1± 4i
Equation
Add 1 to both sides.
The equation (x – 1)2 + 15 = –1 has two solutions, 1 ±
4i.
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Try This One…
Solve 4(x + 3)2 – 10 = –6 for x.
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Thanks for Watching!!!!
Dr. Dambreville