[Music] [Music] hello let's do this journey with me as we face overcome and befriend the challenges ahead we can because we are mathematician i am teacher andrew your soul mate allow me to guide you in learning the different concepts and skills in mathematics 9. when you were in grade 7 you must have learned about linear equations in one variable i have here three bricks and written on each is an equation we will identify whether the equation is linear or not the first brick displays the equation 9x plus 15 equals 91. is it linear let's see if you were correct all right so if you said that it is linear you are correct the second brick displays the equation 2x plus 9 equals 20. is it linear it is also linear now the last brick displays the equation 5x squared minus 23x equals 19. is it linear clearly that isn't a linear equation but do you know why it is because the highest exponent of the variable in the equation is 2 and we call such an equation a quadratic equation in today's episode we should be able to differentiate quadratic equations in one variable from linear equations write such equations in standard form identify the value of the coefficients a b and c and lastly determine whether a situation or word problem illustrates a quadratic equation since we should not vary the skills and concepts we have previously learned we say past not dead to recall a skill you learned when you were in grade 7 let us answer this activity does this task look familiar i hope it does because you were taught by your grade 7 math teacher how to multiply polynomials let us answer item number one in five seconds ready let's do this [Music] and time's up applying the distributive property we should get two x squared minus seven x as the answer we now answer item number two in seven seconds ready let's do this [Music] and time's up here we use the foil method since the given factors are both binomials we shall get x squared plus nine x plus five x plus forty five since there are similar terms we can simplify it into x squared plus fourteen x plus forty five another seven seconds for item number three ready let's do this time is up the answer is two x squared plus five x minus three item number four presents the case square of a binomial for five seconds let's do this [Music] times up the answer is a perfect square trinomial in which the first term is the square of the first term in the binomial the last term is the square of the last term in the binomial and the middle term is twice the product of the terms in the binomial hence the answer is x squared plus 12x plus 36. the last item presents the case sum and difference of two numbers for five seconds let's do this [Music] time is up instead of applying foil method we simply multiply the first terms and then the last terms thus the answer is 25x squared minus 16 a difference of two squares now let's take a look at all the answers in the activity we just did all of them are polynomials of degree two now if we set each equal to zero what do we get quadratic equations in standard form they are in standard form because 1 the terms are arranged in descending powers 2 the polynomial is set equal to 0. in general a quadratic equation in one variable is in standard form if it is of the form ax squared plus bx plus c equals zero where a b and c are real numbers and a should not be equal to zero ax squared is the quadratic term or the term of degree two bx is the linear term or the term of degree one while c is the constant term or the term of degree zero note that a or the leading coefficient should not be equal to zero otherwise the equation will be linear since any quantity multiplied by 0 is equal to 0 and thus can be omitted already [Music] therefore there are quadratic equations where b is equal to zero or there is no linear term they are of the form ax squared plus c equals zero just like 25x squared minus 16 equals zero on the other hand there are those where c is equal to zero or there is no constant term they are of the form a x squared plus b x equals zero just like 2x squared minus 7x equals 0. however the standard form of a quadratic equation is not unique because of the following cases case one considering symmetric property say for instance we are to write the equation five x minus three equals two x squared in standard form one way to do it is to apply the subtraction property of equality so that the left side of the equation contains the quadratic term and the right side is just zero then we arrange the terms in descending powers hence the equation negative 2x squared plus 5x minus 3 equals 0. here the values of the coefficients are as follows a is negative 2 b is 5 and c is negative 3. the other way to write the given equation in standard form is by applying the symmetric property after which we can apply other properties of equality to place all the terms on the left side of the equation and thus the right side is just 0. it so happens that the terms are already arranged in descending powers hence 2x squared minus 5x plus 3 equals 0 is also the standard form of the equation 5x minus 3 equals 2x squared here the values of the coefficients are as follows a is 2 b is negative 5 and c is 3. so the equation 5x minus 3 equals 2x squared can be written in standard form in two ways first is negative two x squared plus five x minus three equals zero and two x squared minus five x plus three equals 0 are considered equivalent quadratic equations case 2 considering greatest common factor of the coefficients our example here is the equation 3x squared plus 9x minus 12 equals 0. although we can see that the equation is already in standard form the coefficients 3 9 and negative 12 which are the value of a b and c respectively are all divisible by three which is their gcf or greatest common factor so dividing both sides of the equation by three we get x squared plus three x minus four equals zero which is also in standard form here the value of the coefficients are as follows a is one b is three and c is negative four notice that their gcf is already one third four they are considered relatively prime so we say that three x squared plus nine x minus twelve equals zero and x squared plus three x minus four equals zero are equivalent quadratic equations but for the purpose of having to think of only one way to write the standard form all answers will be expressed in the form ax squared plus bx plus c equals 0 where the leading coefficient or the value of a must be positive or greater than zero and all nonzero coefficients are just divisible by one how are you holding up i hope everything is fine so to make sure that we are on the same page that we understand what has been discussed let's answer another activity so we say matt the moment [Music] in this activity we have to write the given equation in standard form and identify the value of a b and c again a should not be negative it should be positive or greater than zero and all non-zero coefficients are just divisible by one item number one for 15 seconds let's do this [Music] and it's time to reveal the answers see if you got them all right the standard form is 2x squared minus 2x plus 7 equals 0. the coefficients are as follows a is 2 b is negative 2 and c is 7. let's answer item number 2 for 15 seconds let's do this time up the answers standard form is x squared plus three x minus three equals zero the coefficients a is one b is positive 3 while c is negative 3. item number 3 is next for 30 seconds let's do this [Music] [Music] and exhale here are the answers standard form x squared plus nine x plus 20 equals zero coefficients a is one b is nine while c is twenty we're ready for another one aren't we item number four for 30 seconds let's do this [Music] armpits up check your answers standard form x squared plus x minus two equals zero a is one b is one and c is negative two last item for 30 seconds let's do this [Music] and boom pen or pencil down the answers are the standard form is x squared minus two x minus five equals zero so a is one b is negative 2 and c is negative 5. let's take another look at the answers we expressed the standard form in such a way that the leading coefficient or the value of a is positive and all the non-zero coefficients are relatively prime to enhance our understanding of quadratic equations we will learn how to represent situations using equations and be able to determine whether the situation illustrates a quadratic equation let's read the first sample situation altogether if the square of a number is added to 8 times the number the result is 100 since there is no specified or indicated variable we will just use x square of a number can be translated as x squared added two in symbols is just plus sign eight times the number can be translated as 8x the result is 100 in symbols is equals one hundred hence the situation can be expressed as x squared plus eight x equals one hundred obviously the equation is quadratic since the highest exponent of the variable x is 2. now let's write it in standard form by applying the subtraction property of equality the resulting equation is x squared plus 8x minus 100 equals zero all right it's time to practice the skill we have just learned we say math the moment represent each situation using an equation to determine if it illustrates a quadratic equation [Music] item number one let's read it all together mrs salome charged 3655 pesos worth of groceries on her credit card the balance of her credit card after she made the payment is 2450 pesos for 30 seconds let's do this so and time is up so put your pen down do you think you got it right let's see the situation can be represented by the equation x minus 3655 equals 2450 where x was the balance in the credit card before mrs salume made the payment and since the variable is raised to 1 the equation is not quadratic thus the situation does not illustrate a quadratic equation item number two let's read it first the dimensions of a cube are reduced by five centimeters on each side and the new surface area becomes 48 square centimeters for 30 seconds let's do [Music] this so stop right there whether you did it right or wrong you deserve a hug from yourself now let me discuss this first a cube is a solid figure that is made up of six faces that are all squares the dimensions of a cube are reduced by five centimeters this means that from the original side length which is unknown it is shortened by 5 centimeters thus the change from s to s minus 5. the formula for the surface area of the cube is given by the equation s a equals 6 s squared where s refers to the side length of the cube the new surface area becomes 48 square centimeters in symbols s a equals 48 and 48 must be equal to 6 multiplied by the quantity s minus 5 squared we should realize that both sides of the equation are divisible by six dividing both sides by six we should get eight is equal to the square of the quantity s minus five squaring the binomial on the right side the equation now becomes eight equals s squared minus 10 s plus 25 we see that the highest exponent of the variable is two thus the situation illustrates a quadratic equation combining similar terms and writing the equation in standard form the resulting equation is s squared minus 10 s plus 17 equals zero item number three let's read it all together the area of a rectangular floor is 864 square meters and its perimeter is 120 meters let's do this [Music] so and time is up let me explain and share the answer first let's recall the formula for the area and for the perimeter of rectangle the area is 864 square meters it just means that 864 is the product of the length and the width of the floor the perimeter of the floor is 120 meters this means that twice the sum of the length in the width is 120 and this also means that the sum of the length and the width is 60. say we express the length in terms of the width so we have the equation 60 minus w equals l or l equals 60 minus w now by substituting 60 minus w in place of l in the equation 864 equals l times w it is going to look like this 864 equals 60 minus w times w let's multiply the quantities on the right side so we have 864 equals 60 w minus w squared so clearly the equation is quadratic therefore the situation illustrates a quadratic equation in standard form it can be written as w squared minus 60 w plus 864 equals zero kagamas lest we forget what we have just learned let's have a quick recap through this activity we will call you complete me just supply the missing part of each statement let's have number one a linear equation is of degree one while a blank is of degree two answer quadratic equation statement number two the standard form of a quadratic equation in one variable is blank where a b and c are real numbers and a should not be equal to zero answer a x squared plus b x plus c equals zero third statement when expressed in standard form where a is positive and all the non-zero coefficients are relatively prime the equation 10 equals 4x multiplied by the binomial 2x minus 3 is written as blank and the value of a b and c are blank blank and blank respectively answer four x squared minus six x minus five equals zero a is four b is negative six and c is negative five and that's it kagamath i hope today's episode has been very helpful this is your soul mate teacher andrew but before we go off air let me share one of the mottos i hold into as a mathematics teacher and learner at the same time do not hate what you do not know see you again next time as we face overcome and befriend the challenges ahead we can because we are mathematician [Music] you