Solving Square Roots with Imaginary Solutions

Summary

This 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

00:00

📐 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.

05:02

🔍 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.

Keywords

💡Quadratic Equations

Quadratic equations are polynomial equations of degree two, meaning they have the form ax^2 + bx + c = 0. The video focuses on solving these equations using square roots, which is a method applicable when the discriminant (b^2 - 4ac) is negative, leading to imaginary solutions. For instance, the equation x^2 + 4 = 0 is a quadratic equation that the video demonstrates how to solve.

💡Imaginary Solutions

Imaginary solutions are solutions to equations that involve the square root of a negative number. In the video, imaginary solutions are discussed in the context of quadratic equations that do not have real number solutions. An example given is x^2 = -4, which leads to solutions involving the imaginary unit 'i', where i^2 = -1.

💡Square Roots

The square root operation is used to solve for x in the quadratic equations presented in the video. When taking the square root of both sides of an equation, the video explains that it leads to the absolute value of x, which can be either positive or negative. For example, in the equation x^2 = -4, taking the square root gives the square root of -4, which is 2i.

💡Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering direction. In the context of the video, absolute value is used when taking the square root of x^2, which results in the magnitude of x, not its direction. For instance, in the solution x = ±2i, both 2i and -2i are considered as they are the same distance from zero.

💡Imaginary Numbers

Imaginary numbers are real numbers multiplied by the imaginary unit 'i', which is defined as the square root of -1. The video explains how to handle these numbers when they appear as solutions to quadratic equations, such as in the case of x^2 = -9, where the solution involves 3i.

💡Isolating Variables

Isolating variables is a technique used in algebra to solve equations by getting the variable alone on one side of the equation. The video demonstrates this by isolating x^2 in the equation x^2 + 4 = 0 before solving for x.

💡Discriminant

The discriminant of a quadratic equation (b^2 - 4ac) determines the nature of its solutions. If the discriminant is negative, the equation has complex or imaginary solutions, which is the focus of the video. The script does not explicitly mention the term 'discriminant', but it is the underlying concept when discussing why the equations have imaginary solutions.

💡Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part, typically written as a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the square root of -1. The video discusses solving equations that result in complex numbers, such as x = 4 ± 2√2i.

💡Parentheses

Parentheses are used in algebraic expressions to group terms together. In the video, parentheses are used in expressions like (x - 4)^2 to indicate that the entire term inside the parentheses is squared. The video also shows how to handle equations where the variable is inside parentheses, like 3(x - 4)^2 = -24.

💡Fractions

Fractions are used in algebra to represent parts of a whole. In the video, fractions are present in equations like -1/2(x + 1)^2 = 26, where the fraction is part of the coefficient of the squared term. The video demonstrates how to solve such equations by eliminating the fraction through multiplication.

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