SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE || GRADE 9 MATHEMATICS Q1

Summary

This educational video script focuses on the process of expressing square of trinomials as square of binomials and solving quadratic equations by completing the square. It provides step-by-step examples, including converting non-perfect square trinomials into perfect squares and solving the resulting equations by extracting square roots. The script illustrates various examples, demonstrating how to manipulate algebraic expressions to find the values of x in different quadratic equations.

Takeaways

• 📚 The video explains how to express the square of trinomials as the square of binomials.
• 🔍 It demonstrates solving quadratic equations by completing the square, a method involving adding the square of half the coefficient of 'x' to form a perfect square trinomial.
• 📐 The process involves transforming an equation like 'x^2 + bx' into a perfect square trinomial '(x + b/2)^2'.
• 📘 Examples are provided, such as transforming 'x^2 - 8x + 16' into '(x - 4)^2' and 'x^2 + 3x' into '(x + 3/2)^2'.
• 📙 The video also covers converting perfect square trinomials back into the square of binomials, as shown with 'x^2 + 4x + 4' becoming '(x + 2)^2'.
• 📒 The method is applied to solve quadratic equations, such as 'x^2 + 10x = 3', by creating a perfect square trinomial and then isolating it.
• 📕 After forming the perfect square trinomial, the equation is balanced by adding the constant term to both sides, allowing for the extraction of square roots.
• 📗 The video provides step-by-step solutions for equations, including how to handle non-perfect square roots, like in the equation 'x^2 - 3x = 1'.
• 📖 It explains how to simplify the final answer by combining like terms and rationalizing denominators when necessary.
• 📔 The process is illustrated with multiple examples, each showing a different approach to solving quadratic equations by completing the square.
• 📓 The video concludes with a challenging example, '2x^2 - 3x - 9 = 0', which is solved by adjusting the equation, forming a perfect square trinomial, and then extracting the square root.

Q & A

What is the process of completing the square for a quadratic expression?

Completing the square involves taking a quadratic expression of the form x^2 + bx and adding the square of half the coefficient of x to make it a perfect square trinomial, which is then expressed as (x + b/2)^2.

How do you express a perfect square trinomial as a square of a binomial?

To express a perfect square trinomial as a square of a binomial, you take the square root of the constant term and add it to x, if the middle term is positive, or subtract it from x, if the middle term is negative, resulting in (x ± √constant term)^2.

What is the first step in solving a quadratic equation by completing the square?

The first step is to ensure the quadratic equation is in the form x^2 + bx = c, then add the square of half the coefficient of x to both sides of the equation to form a perfect square trinomial.

How do you find the value of 'b' in the context of completing the square?

The value of 'b' is half the coefficient of the x term in the quadratic expression. For example, if the expression is x^2 - 8x, then b = -8/2 = -4.

What is the purpose of adding and subtracting the same value in the process of completing the square?

Adding and subtracting the same value helps to create a perfect square trinomial on one side of the equation without changing the equation's balance, which is essential for solving the quadratic equation.

Can you provide an example of a quadratic equation solved by completing the square?

An example is the equation x^2 - 8x = 16. By completing the square, it becomes x^2 - 8x + 16 = 16 + 16, which simplifies to (x - 4)^2 = 32, and then solving for x gives x = 4 ± √32.

What is the significance of the term 'perfect square trinomial' in the context of completing the square?

A perfect square trinomial is an expression of the form x^2 + 2bx + b^2, which is the square of the binomial (x + b). It is significant because it allows the quadratic expression to be rewritten in a simpler, squared form that can be easily solved.

How do you determine if a quadratic expression can be expressed as a perfect square trinomial?

A quadratic expression can be expressed as a perfect square trinomial if the coefficient of the x term is an even number, and the constant term is the square of half the coefficient of the x term.

What is the final step in solving a quadratic equation by completing the square?

The final step is to take the square root of both sides of the equation to solve for x, which results in x = ±√(c + square of half the coefficient of x).

Can the method of completing the square be used for any quadratic equation?

The method of completing the square can be used for any quadratic equation, but it is most effective when the quadratic does not factor easily or when the coefficient of the x^2 term is 1.

Outlines

00:00

📚 Introduction to Completing the Square

This paragraph introduces the concept of expressing the square of trinomials as squares of binomials and solving quadratic equations by completing the square. The process involves adding the square of half the coefficient of the linear term to the quadratic expression to form a perfect square trinomial. An example given is 'x squared minus eight x plus sixteen,' which simplifies to '(x - 4) squared.' The paragraph also covers the conversion of non-perfect square trinomials into perfect square trinomials, such as 'x squared plus 3x,' which becomes '(x + 3/2) squared.'

05:07

📘 Examples of Expressing Perfect Square Trinomials

This section provides several examples of expressing perfect square trinomials as squares of binomials. It demonstrates the process for trinomials like 'x squared plus four x plus four,' which simplifies to '(x + 2) squared,' and 'x squared minus 10x plus 25,' which simplifies to '(x - 5) squared.' Each example includes identifying the middle term's coefficient, dividing it by two, and squaring the result to complete the square, ultimately forming a binomial square that matches the original trinomial.

10:09

🔍 Solving Quadratic Equations by Completing the Square

The paragraph illustrates the method of solving quadratic equations by completing the square, using the equation 'x squared plus 10x = 3' as an example. It explains how to adjust the equation to form a perfect square trinomial on the left side, resulting in 'x squared plus 10x + 25,' which simplifies to '(x + 5) squared.' The right side of the equation is then balanced by adding 25, leading to the equation '(x + 5) squared = 28.' The solution involves taking the square root of both sides, yielding 'x + 5 = ±√28,' and then solving for 'x.'

15:11

📐 Advanced Techniques in Completing the Square

This paragraph delves into more complex scenarios of completing the square with equations not in standard form, such as 'x squared minus 3x = 9.' It guides through the process of adjusting the equation to create a perfect square trinomial, resulting in 'x squared minus 3x + 9/4,' which simplifies to '(x - 3/2) squared = 13/4.' The solution process involves taking the square root of both sides and solving for 'x,' considering both the positive and negative roots.

20:12

📉 Solving Challenging Quadratic Equations

The paragraph presents a challenging example of solving a quadratic equation, '2x squared - 3x - 9 = 0,' by first normalizing the coefficients and then completing the square to form 'x squared - 3x/2 + 9/16.' This results in the perfect square trinomial '(x - 3/4) squared = 81/16.' The solution involves extracting the square root and solving the resulting equations 'x - 3/4 = ±9/4,' leading to the final values of 'x.'

25:12

📚 Conclusion on Completing the Square Method

The final paragraph wraps up the discussion on solving quadratic equations by completing the square. It reiterates the process and provides a comprehensive summary of the method, highlighting the importance of forming perfect square trinomials and the steps to solve for 'x' in various scenarios. It also emphasizes the application of this technique in different types of quadratic equations, showcasing its versatility and effectiveness.

Keywords

💡Square of Trinomials

The term 'Square of Trinomials' refers to the algebraic expression that results from squaring a trinomial, which is a polynomial with three terms. In the context of the video, it is used to demonstrate how to express a trinomial's square as a square of binomials, which is a key step in solving quadratic equations by completing the square. For example, the script mentions 'x squared plus bx add the square of half the coefficient of x to make x squared plus b x plus b over 2 squared'.

💡Completing the Square

Completing the square is a method used to solve quadratic equations. It involves manipulating the quadratic equation into a form that allows for easy extraction of the square root. The video script explains this process by adding the square of half the coefficient of the linear term to both sides of the equation, turning it into a perfect square trinomial. An example given is 'x squared plus 10x' where '10 over 2 is equal to five, and after nothing he divides the two, squared not, n so five squared is equal to twenty-five'.

💡Perfect Square Trinomial

A 'Perfect Square Trinomial' is a specific type of trinomial that can be factored into the square of a binomial. It is characterized by having a middle term that is twice the product of the square roots of the first and last terms. The video uses this concept to transform non-perfect square trinomials into perfect ones, as seen in 'x squared plus 3x, divided, three over two squared that is equal to, nine over four'.

💡Quadratic Equation

A 'Quadratic Equation' is a polynomial equation of degree two, typically in the form of ax^2 + bx + c = 0. The video's main theme revolves around solving these equations using the method of completing the square. The script provides several examples of quadratic equations being solved, such as 'x squared plus 10 x is equal to three'.

💡Binomial

A 'Binomial' is an algebraic expression with two terms. In the context of the video, binomials are used in the process of completing the square, where a trinomial is expressed as the square of a binomial. For instance, 'x plus 3 over 2 squared' is a binomial squared, as explained in the script.

💡Coefficient

The 'Coefficient' is a numerical factor that multiplies a variable in an algebraic expression. In the script, coefficients are used to determine the terms that will be squared to form a perfect square trinomial, such as 'the coefficient of x' which is used to calculate 'b over 2' in 'x squared plus bx'.

💡Square Root

The 'Square Root' operation is used to find a number that, when multiplied by itself, gives the original number. In the video, square roots are extracted to solve the quadratic equations after they have been transformed into perfect square trinomials. An example is 'x plus five is equal to positive or negative square root of twenty eight'.

💡Algebraic Manipulation

Algebraic Manipulation refers to the process of changing the form of an equation or expression using algebraic rules without changing its value. The video demonstrates this by adding, subtracting, and rearranging terms to complete the square, as shown in 'x squared plus 10 x plus 25 so it only on a perfect square trinomials now'.

💡Linear Term

A 'Linear Term' in a polynomial is the term with the variable raised to the power of one. In the script, the linear term is associated with the coefficient 'b' in the quadratic equation, which is manipulated to form a perfect square trinomial, as in 'x squared plus b x'.

💡Constant Term

The 'Constant Term' in a polynomial is the term without any variables. It is the term that, when squared, becomes the last term in a perfect square trinomial. The script mentions this in the context of completing the square, such as 'the last term or constant term you know, minus 4 squared'.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of its factors. In the video, factoring is used to express a trinomial as a square of a binomial, which is a form of factoring. For example, 'x squared plus four x plus four' is factored as '(x + 2) squared'.

Highlights

Introduction to expressing square of trinomials as square of binomials.

Explanation of completing the square to solve quadratic equations.

Method to create a perfect square trinomial by adding the square of half the coefficient of x.

Example of transforming x squared minus 8x into a perfect square trinomial.

Technique for converting non-perfect square trinomials into perfect square trinomials.

Demonstration of expressing x squared plus 3x as a square of binomial.

Process of solving quadratic equations by completing the square with given examples.

Step-by-step guide to completing the square for the equation x squared plus 10x equals 3.

Conversion of perfect square trinomials into square of binomials with multiple examples.

Solving quadratic equations by extracting square roots after completing the square.

Example of solving x squared minus 3x equals 1 using the completing the square method.

Explanation of how to handle equations not in standard form when completing the square.

Final answer presentation for the equation x squared plus 10x equals 3 using completing the square.

Process of solving the challenging equation 2x squared minus 3x minus 9 equals 0 by completing the square.

Detailed walkthrough of dividing all terms by the leading coefficient to simplify the equation.

Final solutions for the equation with steps to arrive at x equals three and x equals negative three over two.

Summary of the method to solve quadratic equations by completing the square with practical examples.

Transcripts

hello kawamat in this video

we express square of trinomials as a

square of binomials

and we solve quadratic equation by

completing the square

completing the square so this

is to complete the square of the

expression

x squared plus bx add the square

of half the coefficient of x to make

x squared plus b x plus b over 2 squared

okay so equation

no parameter perfect square trinomial

is an expression so para dagdagan

an x squared plus b x and gaga v nathan

connie nothing in volume v over 2

tables squared langnatan

halimbawa x squared

minus eight x plus si expression

and that is negative eight and then

indeed divide that into two

so negative eight divide two is equal to

negative four

package

so negative eight divided by two

the answer is negative four after

that squared langnakuha

negative four so negative four

squared is equal to sixteen

so parasan button sixteen at all so it

don't see sixteen

x squared minus eight x

sorry

x squared minus eight x

again x squared minus eight

x plus sixteen okay

so e big sub n e to i is a non

perfect squared trinomials so panoramic

is a perfect square trinomials

the last term or constant term you know

minus 4 squared

next

we have x squared plus 3x

divided

three over two squared that is equal to

nine over four bucket three times

three that is nine two times two that is

four so again

x squared plus three x

plus nine over four so it

is a now perfect square trinomial

square of binomial

so x plus 3 over 2 squared

so going perfect square trinomial

at from perfect square trinomial going

square

of binomial so i'll give you more

examples

express each of the following perfect

square trinomials

as a square of binomial first

x squared plus four x plus four

so it'll given a perfect square

trinomials

nothing square up a binomial

root 4 that is 2 so therefore

a square of binomial is x

plus 2 squared so

now it

x squared plus 12 x plus 36

again last

part or your last and last term yo

so square root 36 is six so therefore

the square of binomial is

x plus six squared

another we have x squared

minus 10 x plus 25

so let another square root of 25

5 so i don't sign negative

so therefore the square of binomial is

x minus 5 squared

last example x squared

minus five x plus 25

over four so nothing

squared and that is

five over two square root of 25 is 5

square root of 4 is 2 so that is 5 over

2

so therefore our square of binomial is

x minus 5

over 2 squared

the quadratic equation by completing the

square

okay so simulating

given the quadratic equation x squared

plus 10 x is equal to three

so capacity solved diode and quadratic

equation gamma

and completing the square

okay you constant term not necessary

side and then you form the x squared

plus b x or your quadratic at linear

term nothing

that's a left side now equation

okay x squared plus 10

x plus

so that is 10 over 2 is equal to five

and after nothing he divides the two

squared not

n so five squared is equal to

twenty-five so alumni

25 squared

trinomial so your expression not n i

x squared plus 10 x

plus 25 so it only on a perfect square

trinomials now

so kapaki no are nothing in square of

binomials that is

x plus five squared

okay after

perfect square trinomials no we need to

add

25 to both sides of the equation

and 25

x squared plus 10 x

plus 25 is equal to 3

plus 25 so after nyan and

express the perfect square

trinomial so your pst is perfect square

trinomial on the left side

left side a square

of binomial so that will be x

plus 5 squared

is equal to 28.

during solving quadratic equation by

extracting the square root

okay so extract nothing

so solve the resulting quadratic

equation by extracting the square root

so

x plus five is equal to positive or

negative square root of twenty eight

so thinking about

so remember class 28

and number 28 i made on factor not my

perfect squared

so independent

[Music]

squared that is four

and seven so therefore the square root

of four times seven it will become two

square root of seven because see perfect

squared in four not

n so magnitude equation at n

x plus five is equal to

positive or negative two square root of

seven neon predicting

solving equation melon equation

x plus pi is equal to two square root

of seven atom parallel equation at the

end

x plus five is equal to negative two

square root of seven

okay solve nothing equation not n

so first x plus five

is equal to two square root of seven so

i don't

know but like nothing c fives are

right side so muggy king

x is equal to negative five

plus two square root of seven and this

is now the final answer hindi pretty

nothing it

uh simplify knowing the party adds

negative five plus two square root of

seven

then basila c2 i make a sum and square

root of seven

si pi voila so this is now the final

answer

in the muppet department negative 5

at positive 2 next

okay in the next equation we have x plus

5

is equal to negative two square root of

seven

so same process lipitlan not in c

positive five

suckabiling side so much again

x is equal to negative five

minus two square root of seven

okay

another example

x squared minus three x is equal to one

so again bhagavatayama proceeds the next

step

i cook perfect square

trinomial

so negative 3

over 2 after that squared not

n so negative three over two

squared so negative three times negative

three that is positive nine

and two times two is positive four so he

picks

is nine over four so that will become

x squared minus three x plus

nine over four at

that will become x minus 3 over 2

squared important

square of binomial

in both equations nine

over four so making x squared

minus three x plus nine over four

is equal to one plus nine over four

next express the perfect square

trinomial on the left side

of the equation as a square of binomial

so since kanina ke nuhan and nothing in

square

x minus 3 squared is equal to 13 over 4

so pan cooling in 13 over 4

four times one so that is four plus

nine thirteen over four

okay so i had 13 over four

and um

square root so therefore making

x minus 3 over 2

is equal to positive or negative the

square root of 13

over 4. so again

hindi paito no bucket value 4 is a

perfect squared

so maggi king and again x minus 3 over 2

is equal to positive or negative

the square root of 13 over 2.

so in the piano independent final answer

question

so we need to solve the equation

in the next slide so it in the lower

equation nothing

okay sold nothing in the lower equation

i am

first is the x minus three over two

is equal to square root of thirteen over

two

so another we not in dito lipatna

c negative three over two sub capillary

sides of magicking and yan

equal to three over two plus square root

of thirteen over two

okay the subfraction capacitor

plus square root of thirteen over

two okay

ganon didn't process a cabin side

x minus three over two is equal to

negative square root of thirteen over

two

so leap at nothing see negative three

over two then the left side

so anomaly area three over two

minus square root of thirteen over two

so the value of x

is x is equal to three

minus square root of thirteen over two

okay i hope nasu sundan union processor

how to solve quadratic equation by

completing the square

so i'll give you another example

okay your equation not in a standard

form

nothing

and then squared so 5 over 2 squared is

equal to 25

over 4. so a magnetic expression and

nothing is

x squared plus five x

plus 25 over four

tatandano and processor okay next

so to get the square of binomial so

so that is five over two so x plus

five over two squared so malignant

i don't say equation una

we need to add 25 over

four sun began 25 over four dito

the same squared yo

both side so making x squared plus

5x plus 25 over 4

is equal to negative 4 plus 25

over 4 and then express

the perfect square trinomial as

square

so that is x plus 5 over 2 squared

is equal to 9 over 4 and then

extract the square root so therefore

it will become x plus five over two

is equal to positive or negative

square root of nine over four but c nine

zero

c four is a perfect squared so a normal

gigging equation at them

that will be com x plus

five over two is equal to positive or

negative three

over two so again maritime the long

equation

we need to solve the value of x

so solve nothing equation

x plus five over two is equal to three

over two so on gaga when lipid nut is c

five over two

to solve for x suckability side so

magicking and again

negative five over two plus three over

two

so madali lang is sold as a same demand

and denominator

so we can add the numerator so negative

five plus three

that is negative two over two

so same on denominator

so therefore your value of x

is negative one okay in the other side

we have x plus five over two

is equal to negative three over two

so leave at length pipe over to right

side

and still become negative five over two

minus three over two then add nothing

negative five minus three is negative

eight

over two and negative eight over two

your x

is negative four okay

so last

challenging last example

two x squared minus three x minus nine

equals zero

so on in the observed muscle given

equation not

n so you numerical coefficient that is a

quadratic term

you need to okay

so 2x squared minus 3x is equal to 9

and then it did divide not in

all the terms by 2 so mug again

so 2x squared divided so x squared long

minus three x over two is equal to nine

over two

so after nitto and the n going out in

perfect square trinomials

multiplication so negative three

over two times one half the answer is

negative three over four so after n

squared naught is

negative three over four the answer is

nine over sixteen so c nine over sixteen

yang ida

so therefore your equation will become

x squared no your expression rather

x squared minus three x over two plus

9 over 16 and then the square of

binomial is

x minus 3 over 4 squared

so alumni not n so after dividing the

terms by 2 and getting the perfect

square trinomials

so add the latency 9 over 16

to both sides of the equation so

x squared minus 3x

plus over 2 plus 9 over 16

equals nine over two plus nine over

sixteen

and then going nothing to square up

binomial and that is x

minus three over four squared d to the

month

lcd lcd dito 16 so 16 divide 2 the

answer is

8 times 9 that is 72

16 divided by 16 1 times nine is nine so

72

plus nine the answer is 81 k negan 81

over 16

and then extract the square root so x

minus 3

over for x minus 3 4 is equal to

positive

or negative the square root of 81 over

16.

then i know we know we all know that 81

and 16 is a perfect square

so therefore x minus 3 over 4 is equal

to positive

or negative nine over four so solve that

two equation

so solve nothing in the equation unamuna

opacity

x minus three fourth is equal to

3 4 plus 9 over 4 is 12

over 4 question a similar fraction a big

sub

n but it has a denominator so add

nothing

numerator and then 12 divide four

the value of x is three so cabinet side

x minus three over four equals negative

nine over four

so lipa tell it nothing to see negative

three four making

positive three over four minus nine over

four

so three minus nine that is negative six

over four and then pretty nothing

lowest term yeah so the lowest term of

negative six

over four is negative three over two