Solving Square Roots with Imaginary Solutions
Summary
TLDRThis video tutorial teaches solving quadratic equations with imaginary solutions using square roots. It covers isolating x-squared, applying square roots to both sides, and handling absolute values. Examples include equations like x^2 + 4 = 0 leading to solutions like ±2i, and more complex scenarios like (x-4)^2 = -8, resulting in x = 4 ± 2√2i. The video also addresses equations with variables inside parentheses and fractions, emphasizing the importance of order in writing solutions.
Takeaways
- 🔢 Solving quadratic equations with imaginary solutions involves taking square roots of negative numbers.
- 📐 When isolating x^2 and taking the square root, the result represents the absolute value of x.
- 🧮 The square root of a negative number can be broken down into i (the square root of -1) multiplied by the square root of the positive number.
- 📉 For equations like x^2 = -4, the solution is x = ± 2i, indicating both positive and negative imaginary numbers.
- 🔄 When dealing with equations like 3x^2 = 27, dividing by the coefficient of x^2 isolates the term before taking the square root.
- 📌 The absolute value notation indicates that x could be either positive or negative, so solutions are given as x = ± √-9.
- 🤔 For equations where x is inside parentheses, such as (x-4)^2 = -8, first isolate the squared term and then solve for x.
- 🔄 When solving equations with fractions or complex coefficients, like (-1/2x + 1)^2 = 25, simplify the equation before taking the square root.
- 📘 The order of operations for writing solutions is: numbers first, then letters, and finally square roots.
- 🔗 When an equation has x inside a squared term, you must consider both positive and negative square roots to find the complete solution.
Q & A
What is the first step when solving a quadratic equation with imaginary solutions?
-The first step is to isolate the x squared term, ensuring there are no x terms left on the equation.
How do you solve the equation x squared plus 4 equals 0?
-You subtract 4 from both sides to get x squared equals negative 4, then take the square root of both sides to find x equals positive or negative 2i.
What does the square root of a negative number represent?
-The square root of a negative number represents an imaginary number, where the square root of negative one is denoted by 'i'.
How do you handle the absolute value when solving quadratic equations with imaginary solutions?
-The absolute value signifies the distance from zero, which means the solution could be either positive or negative, representing both possibilities.
What is the solution to the equation 3x squared equals 27?
-After dividing both sides by 3, you get x squared equals negative nine, and taking the square root gives x equals positive or negative 3i.
Can you provide an example of breaking down a square root into simpler components?
-Yes, for the equation x squared equals negative 32, you can break down the square root of 32 into 4 times the square root of 2 times negative one, which simplifies to 4i.
How do you solve equations where x is inside a set of parentheses?
-First, you isolate the squared term, then take the square root of both sides, and finally solve for x by considering both positive and negative possibilities.
What is the process for solving equations with complex coefficients like in the example 'three times (x minus four) squared equals negative twenty-four'?
-You divide by the coefficient to isolate the squared term, take the square root of both sides, consider both positive and negative solutions due to the absolute value, and then solve for x.
How do you handle fractions in quadratic equations?
-You can either multiply by the denominator to clear the fraction or simplify the equation by subtracting a constant to make it easier to solve.
What is the final step when solving for x in the equation 'negative 1/2 x plus 1 squared equals 26'?
-After taking the square root and simplifying, you subtract 1 from both sides to solve for x, resulting in x equals negative one plus or minus 5i root 2.
Why is it important to put the number before the plus or minus sign when solving for x in equations with absolute values?
-Placing the number before the plus or minus sign ensures that the number is correctly associated with both the positive and negative possibilities of the solution.
Outlines
📐 Solving Quadratic Equations with Imaginary Solutions
This paragraph explains how to solve quadratic equations that result in imaginary solutions using square roots. It begins with the equation x squared plus 4 equals 0, demonstrating how to isolate x squared and then take the square root of both sides to find the solution x equals positive or negative 2i. The explanation continues with another example, negative 3x squared minus 27 equals 0, showing how to isolate x squared and solve for x equals positive or negative 3i. The paragraph also covers how to handle equations where the variable is inside a set of parentheses, such as 3 times (x minus 4) squared equals negative twenty-four, and how to deal with fractions and more complex equations like negative 1/2 x plus 1 squared equals 26. Each step is detailed, explaining the process of taking square roots and dealing with imaginary numbers.
🔍 Detailed Steps for Solving Complex Quadratic Equations
The second paragraph delves into solving more complex quadratic equations with imaginary solutions. It starts with the equation 3 times (x minus 4) squared equals negative twenty-four, showing how to simplify and solve for x equals 4 plus or minus 2i root 2. The paragraph then tackles an equation involving fractions, negative 1/2 x plus 1 squared equals 26, and demonstrates the process of simplifying and solving for x equals negative one plus or minus 5i radical 2. The focus is on breaking down complex equations into manageable steps, explaining the importance of order when dealing with numbers, letters, and square roots, and ensuring that the solution accounts for both positive and negative possibilities due to the absolute value.
Mindmap
Keywords
💡Quadratic Equations
💡Imaginary Solutions
💡Square Roots
💡Absolute Value
💡Imaginary Numbers
💡Isolating Variables
💡Discriminant
💡Complex Numbers
💡Parentheses
💡Fractions
Highlights
Introduction to solving quadratic equations with imaginary solutions.
Isolating x squared in the equation x squared plus 4 equals 0.
Deriving the solution x squared equals negative 4 by subtracting 4 from both sides.
Taking the square root of both sides to find the absolute value of x equals the square root of negative 4.
Breaking down the square root of negative 4 into the square root of negative 1 and the square root of 4.
Simplifying to find that x equals positive or negative 2i.
Isolating 3x squared in the equation negative 3x squared minus 27 equals 0.
Solving for x squared equals negative nine by dividing both sides by negative three.
Taking the square root of both sides to find the absolute value of x equals the square root of negative nine.
Expressing the square root of negative nine as 3i.
Solving for x equals positive or negative 3i.
Isolating x squared in the equation x squared equals negative 32.
Taking the square root of both sides to find the absolute value of x equals the square root of negative 32.
Breaking down the square root of negative 32 into the square root of negative 1, the square root of 32, and further into the square root of 16 and the square root of 2.
Solving for x equals positive or negative 4 root 2i.
Handling equations where x is trapped inside parentheses, such as 3 times (x minus 4) squared equals negative twenty-four.
Dividing by three to isolate (x minus 4) squared equals negative eight.
Taking the square root of both sides to find the absolute value of (x minus 4) equals the square root of negative eight.
Solving for x equals 4 plus or minus 2i root 2.
Addressing equations with fractions and parentheses, such as negative 1/2 (x plus 1) squared equals 26.
Multiplying by negative 2 to simplify the equation to (x plus 1) squared equals negative 50.
Taking the square root of both sides to find the absolute value of (x plus 1) equals the square root of negative 50.
Solving for x equals negative one plus or minus 5i root 2.
Transcripts
we're going to solve quadratic equations
using square roots but these quadratic
equations will have imaginary solutions
so hopefully you've watched the video or
at least have learned somewhere what
imaginary solutions are what imaginary
numbers are so these are going to be
equations that end up having solutions
that are imaginary numbers
all right let's hop right into it if we
were to solve the equation x squared
plus 4 equals 0 we can start by
isolating the x squared because there's
no X terms all right so if we were to
subtract 4 from both sides we get x
squared equals negative 4 take the
square root of both sides we're going to
get the square root of x squared equals
the square root of negative 4 this side
right here will give you the absolute
value of X because by definition when
you square an unknown and then square
root you get the absolute value equals
square root of negative 4 we can break
up the square root of negative 4 into
the square root of negative 1 and the
square root of 4 and the square root of
negative 1 is equal to I the square root
of 4 is 2 so these two things multiplied
together so this would be 2i square root
or the absolute value means distance
from 0 so X is equal to a distance of 2i
from 0 so you could either be 2i
to the right of 0 or to the left of 0
meaning this could be positive or
negative 2i so this is X is equal to
positive or negative 2i you could also
write it like this
x equals 2 i or x equals negative 2 it
means the same thing positive negative
means both positive and negative or you
could write them separately like this
that's totally fine ok let's take a look
at another example of equation that has
an I might I'm gonna get rid of that
title that's fine an equation that has
an imaginary solution so if we look at
negative 3x squared minus 27 is equal to
0 okay we want to isolate the X cursor
and bring the 27 over to the other side
by adding it so you end up getting 3x
squared is equal to positive 27 divide
both sides by negative
three so you get x squared is equal to
negative nine take the square root of
both sides square root of x squared
equals the square root of negative nine
all right this gives us absolute value
of x this gives us there's two different
numbers multiplied together that give
you negative 9 1 is negative 1 and then
1 is 9 this turns into an eye this turns
into a 3 so absolute value of X is equal
to 3 I now because there's absolute
values which means this could be a
positive or a negative we don't know
which one and we have to account for
both of them X is equal to positive or
negative 3 guy you could also say x
equals 3 I or x equals negative 3i I'm
going to use this notation going for
just 4 it's quicker but they both they
mean the same thing ok let's look at
this example here we have x squared
equals negative 32 guys are going to
square root of both sides it's already
nice and isolated for us so negative
square root of 32 this side is easy it's
absolute value of x okay this sign we
have a couple of things right we can do
square root of negative 1 and the square
root of 32 we can break 32 up so 32 is
divisible by a perfect square 32 is
divisible by 16 and 16 is a perfect
square over here and then this equals I
this is root 2 and then this is 4 all
right
so these all multiply to negative 30
that's your kind of quick check to make
sure that everything's okay 16 times 2
times negative 1 equals negative 32 so
all these things are fine this is equal
to I this you can't do anything with and
this is 4 in putting these in the order
so the number comes first a nice number
comes first the letters come second and
then the square roots always come last
okay
so to get the X out of the absolute
values we understand that the X could
have originally been positive this or
negative this so X is equal to either
positive or negative 4
route 2 and there is our solution okay
we're going to do two more now what
happens when we have square root
equations that the X is trapped inside
of a set of parentheses or something
like this three times X minus four
squared is equal to negative twenty-four
these ones are probably the longest
alright first we got to divide by three
so get the X minus four squared is equal
to negative eight now I'm going to take
the square root of both sides I'm going
to simplify this a little bit so this is
the absolute value of x minus four is
equal to square root of negative 8 so
the square root of negative eight
we can break up into the square root of
negative 1 times the square root of 8
but 8 we can break up further into the
square root of 4 which is a perfect
square square root of 2 which is not so
here are our three things
this becomes an I this becomes a 2 and
this becomes a root 2 ok so we have that
X minus 4 is equal to positive or
negative number comes first two then the
letter I and then the radical root 2
positive or negative because of the
absolute value and we need to get the 4
on to the other side this is not solved
yet so if we add 4 to the other side
we're going 4 plus or minus 2i root 2 it
is very important that you put this
number whatever you see here put it in
front of the plus or minus because we
don't want to put it afterwards it's not
being added and subtracted this is only
positive so it's got to go in front of
this and this is our solution X is equal
to 4 plus or minus 2i root 2 ok we'll do
one more let's do ooh
craziness fractions everybody's favorite
negative 1/2 X plus 1 quantity squared
will do plus 1 equals 26 okay it's a
first step you could either multiply by
negative 2 or subtract one subtracting
one of things a little bit easier so
we're going to do that negative 1/2 X
plus 1 squared equals 25 then we're
going to multiply this side by negative
2 so when you multiply this by negative
2 negative two times that going 1/2 it's
just 1 we're gonna multiply this ever
negative 2 as well so
X plus 1 quantity squared is equal to
negative 50 okie-dokie and take the
square root of both sides we're gonna
get the absolute value of x plus 1 on
this side
equals square root of negative 50 over
here we're going to separate the
negative 1 from the 50 okay and it's
supposed to be 50 each 50 as is
divisible by a perfect square so 25 is a
perfect square and 25 times 2 is 50 just
carry over that negative 1 so square
root of negative 1 is I square root of
25 is 5 and the square root of 2 we're
just gonna leave there so X let's just
get rid of that because we know that
it's going to be to get rid of the
absolute value we're gonna have to have
a positive negative over here so X plus
1 equals positive or negative number
comes first five letter comes second I
and then square root comes last two and
then we gotta get rid of the 1 so we're
going to subtract it from both sides
it's gonna be X is equal to when we
subtract it's going to be negative one
but again Asko in front of the plus or
minus plus or minus 5i radical 2 there's
our answer
that's it
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