SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE || GRADE 9 MATHEMATICS Q1
Summary
TLDRThis 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
📚 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.'
📘 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.
🔍 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.'
📐 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.
📉 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.'
📚 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.
Mindmap
Keywords
💡Square of Trinomials
💡Completing the Square
💡Perfect Square Trinomial
💡Quadratic Equation
💡Binomial
💡Coefficient
💡Square Root
💡Algebraic Manipulation
💡Linear Term
💡Constant Term
💡Factoring
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
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