In previous article of Algebra I, we had already explained the fundamentals of Rational expressions and polynomials, today in this article we are going to continue with another topic of algebra 1 that is Quadratic equations and quadratic functions. The introductory part of quadratic equation was learned by students in their previous classes but still to make you guys remember it, let us start with the basics concept of Quadratic equations.
As we all know that every algebraic equation is a polynomial function and so on Quadratic equation is also, but the point is that every quadratic equation is a non-linear polynomial function of second order and it is in the form of monomial because quadratic equation has many derivatives but all of them are of a single variable, Suppose x is an unknown variable then its quadratic equation could be of the form:
Ax2 + Bx + C = 0, here A, B, and C are integer constants. One compulsion to have this form as quadratic is that “A is not equal to zero”, if it does then the equation is said to be linear not quadratic.
For example:x2 + 2x = 4 is said to be quadratic because it can be written in the form of Ax2 + Bx + C = 0,
Let us take another example;
Suppose given equation is as 2x2 – 1/x + 2 = 0
Here 1/x can be rewritten as x-1, and quadratic can’t have x raised to the power other then 1 or 2. So the equation is not quadratic.
Now let us come to the main point and that is “ how to solve any quadratic equation?”, in previous article you guys have learned the factoring procedure and in between that session it was nicely explained that factoring can be used to solve any algebraic equation, so quadratic equation is also an algebraic form. Let’s see how factoring is being applied to evaluate quadratic equation solutions.
There are 3 simple steps need to be executed to sort out the quadratic equation:
1. First, write the given equation into standard form and for this you can use distributive property and additional property to combine like terms and put them into one side of ‘=’ sign.
2. Now factor the obtained equation by using simple factoring technique.
3. Now evaluate the possible values of variable by putting each factor equals to zero according to Zero product law.
Let us take an example and observe the actual execution of above defined steps:
(6x + 8)x = 14
Apply distributive property
6x2 + 8x = 14
Imply addition property
6x2 + 8x – 14 = 0 ( in standard form)
Now apply the factoring technique as:
6x2 – 14x + 6x -14 = 0
Now take the common factor out
6x( x-1) – 14 ( x-1) = 0
(x - 1) (-6x + 14) = 0
Use the zero product law
(x - 1) = 0 or (-6x + 14) = 0
x = 1 or x = -14/6 = -7/3
This is the specified way to sort out quadratic equation by using factorization. There is one more way to sort out these types of equations and that is Extracting square roots. After extracting square roots equation automatically comes into the correct form, for example:
( 2x-5 )2 + 5 = 3
Use additional; property to add -5 both sides of the ‘=’ sign
( 2x-5 )2 = -2
Now extract square roots
±√(2x-5)2 = ±√(-2) (negative and positive sign both are included when square root of any value is taken)
(2x – 5) = ±i√ 2 ( simplify the radical and apply the definition of ‘I’)
Apply addition property of equality
2x = 5 + ±i√ 2
Now use division property of equality
x = (5 + ±i√ 2)/2
There are two more ways to evaluate any quadratic equation, one of them is “completing the square” and other is “ quadratic formula”. We will discuss both of them but let us start with Completing the square method.
Let’s take an example and in parallel of its solution we will explain the implementation of this method,
Given equation is as: 3x2 + 4x – 7 = 0
First step: use addition equality property and isolate the variable terms like x2 and x, at one side of the equation.
3x2 + 4x = 7
Second step: use division equality property and divide each term on both sides by integer coefficient of term ‘x2’.
3x2/ 3 + 4x/3 = 7/3
x2 + (4/3)x = 7/3
Third step: now take coefficient of term x, square it and add in both sides of equation; here X’s coefficient is 4/3 so its ½ is 2/3, now square of (2/3) is added to both sides of the equation:
x2 + (4/3)x + (2/3)2 = 7/3 + (2/3)2
x2+ (4/3)x + (4/9) = 7/3 + (4/9)
Multiply 7/3 with 3/3 to have the same denominator on RHS
x2 + (4/3)x + (4/9) = 21/9 + (4/9)
x2 + (4/3)x + (4/9) = 25/9
Next step: factor the left side by using the standard formula:
( a2 + 2ab + b2) = (a + b )2
So , x2 + (4/3)x + (4/9) = 25/9
(x + 2/3 )2 = 25/9
Next step: use; extract the square root technique for finding the relative roots of the equation
√(x + 2/3 )2 =±√( 25/9)
x + 2/3 = ± 5/3
Again use addition property of equality
x = -2/3 ± 5/3
Now two equations can be formed as
x = -2/3 + 5/3 OR x = -2/3 - 5/3
By solving these equations
x = 1 or x = -7/3
This method is used to solve complex problems and for some easy problems prefer the previous method, which is much easier than this one.
Now it’s time to talk about the standard quadratic formula and its use to evaluate Quadratic function. If quadratic equation is in the form of
Ax2 + Bx + C = 0 then the standard quadratic formula is represented as:
; here (b2 – 4ac) is called determinant of the function, which decides the behaviors of the roots of quadratic equation.
While using this formula, the only task for students is to convert the given equation into standard form, for example:
Given equation is: (y + 3)2 = ( y – 2)
Rewrite the above equation as its factors:
( y + 3) ( y + 3) = (y - 2)
Now use the standard distributive property to combine like terms:
y2 + 6y + 9 = y – 2
Use addition equality property:
y2 + 6y + 9 – y + 2 = 0
y2 + 5y + 11 = 0
Now the equation is in standard form so we can implement the quadratic formula, first grab the values of a, b and c from the equation.
a = 1, b = 5 and c = 11.
Put this value in the formula
; by using the ‘i’ definition roots can be easily determined.
This is all about Quadratic equations and functions. In this article we have explored all the possible ways of solving any quadratic equation. These four ways of evaluating quadratic equation covers the topic of quadratic function in grade XII math. Students of this class most of the times fixed in troubles because there are several ways in which a similar problem can be asked and every time there is a bit of changes in the solution process, so it is hard to memorize all the ways and forms of quadratic equation. This task only can be accomplished by doing practice as much they can and to help students in it Online math tutor websites do their role by giving immediate assistance and various worksheets to solve. If any kind of doubt stays in student's mind related to any topic then the whole fundamentals gets messed up, so it must be required that student asks his queries on regular basis and this will happen only when student does not hesitate in front of tutor which most often happen. To troubleshoot this problem, Online services uses online Chatting option via text or video, to build a better conversation between students and tutors, and because most of us are friendly with Internet platform, the access of this service gets easier.
In grade XII math content Algebra is covering most of the problem region so it gets more essential to learn its each and every problem with all the possible ways of query presentation. For having the detailed descriptions of other topics of Grade XII math syllabus, keep following the successive articles that will surely add a wide range of knowledge.
In upcoming posts we will discuss about Exponents in on Grade XII and Methods of data representation. Visit our website for information on Maharashtra state board
As we all know that every algebraic equation is a polynomial function and so on Quadratic equation is also, but the point is that every quadratic equation is a non-linear polynomial function of second order and it is in the form of monomial because quadratic equation has many derivatives but all of them are of a single variable, Suppose x is an unknown variable then its quadratic equation could be of the form:
Ax2 + Bx + C = 0, here A, B, and C are integer constants. One compulsion to have this form as quadratic is that “A is not equal to zero”, if it does then the equation is said to be linear not quadratic.
For example:x2 + 2x = 4 is said to be quadratic because it can be written in the form of Ax2 + Bx + C = 0,
Let us take another example;
Suppose given equation is as 2x2 – 1/x + 2 = 0
Here 1/x can be rewritten as x-1, and quadratic can’t have x raised to the power other then 1 or 2. So the equation is not quadratic.
Now let us come to the main point and that is “ how to solve any quadratic equation?”, in previous article you guys have learned the factoring procedure and in between that session it was nicely explained that factoring can be used to solve any algebraic equation, so quadratic equation is also an algebraic form. Let’s see how factoring is being applied to evaluate quadratic equation solutions.
There are 3 simple steps need to be executed to sort out the quadratic equation:
1. First, write the given equation into standard form and for this you can use distributive property and additional property to combine like terms and put them into one side of ‘=’ sign.
2. Now factor the obtained equation by using simple factoring technique.
3. Now evaluate the possible values of variable by putting each factor equals to zero according to Zero product law.
Let us take an example and observe the actual execution of above defined steps:
(6x + 8)x = 14
Apply distributive property
6x2 + 8x = 14
Imply addition property
6x2 + 8x – 14 = 0 ( in standard form)
Now apply the factoring technique as:
6x2 – 14x + 6x -14 = 0
Now take the common factor out
6x( x-1) – 14 ( x-1) = 0
(x - 1) (-6x + 14) = 0
Use the zero product law
(x - 1) = 0 or (-6x + 14) = 0
x = 1 or x = -14/6 = -7/3
This is the specified way to sort out quadratic equation by using factorization. There is one more way to sort out these types of equations and that is Extracting square roots. After extracting square roots equation automatically comes into the correct form, for example:
( 2x-5 )2 + 5 = 3
Use additional; property to add -5 both sides of the ‘=’ sign
( 2x-5 )2 = -2
Now extract square roots
±√(2x-5)2 = ±√(-2) (negative and positive sign both are included when square root of any value is taken)
(2x – 5) = ±i√ 2 ( simplify the radical and apply the definition of ‘I’)
Apply addition property of equality
2x = 5 + ±i√ 2
Now use division property of equality
x = (5 + ±i√ 2)/2
There are two more ways to evaluate any quadratic equation, one of them is “completing the square” and other is “ quadratic formula”. We will discuss both of them but let us start with Completing the square method.
Let’s take an example and in parallel of its solution we will explain the implementation of this method,
Given equation is as: 3x2 + 4x – 7 = 0
First step: use addition equality property and isolate the variable terms like x2 and x, at one side of the equation.
3x2 + 4x = 7
Second step: use division equality property and divide each term on both sides by integer coefficient of term ‘x2’.
3x2/ 3 + 4x/3 = 7/3
x2 + (4/3)x = 7/3
Third step: now take coefficient of term x, square it and add in both sides of equation; here X’s coefficient is 4/3 so its ½ is 2/3, now square of (2/3) is added to both sides of the equation:
x2 + (4/3)x + (2/3)2 = 7/3 + (2/3)2
x2+ (4/3)x + (4/9) = 7/3 + (4/9)
Multiply 7/3 with 3/3 to have the same denominator on RHS
x2 + (4/3)x + (4/9) = 21/9 + (4/9)
x2 + (4/3)x + (4/9) = 25/9
Next step: factor the left side by using the standard formula:
( a2 + 2ab + b2) = (a + b )2
So , x2 + (4/3)x + (4/9) = 25/9
(x + 2/3 )2 = 25/9
Next step: use; extract the square root technique for finding the relative roots of the equation
√(x + 2/3 )2 =±√( 25/9)
x + 2/3 = ± 5/3
Again use addition property of equality
x = -2/3 ± 5/3
Now two equations can be formed as
x = -2/3 + 5/3 OR x = -2/3 - 5/3
By solving these equations
x = 1 or x = -7/3
This method is used to solve complex problems and for some easy problems prefer the previous method, which is much easier than this one.
Now it’s time to talk about the standard quadratic formula and its use to evaluate Quadratic function. If quadratic equation is in the form of
Ax2 + Bx + C = 0 then the standard quadratic formula is represented as:
; here (b2 – 4ac) is called determinant of the function, which decides the behaviors of the roots of quadratic equation.While using this formula, the only task for students is to convert the given equation into standard form, for example:
Given equation is: (y + 3)2 = ( y – 2)
Rewrite the above equation as its factors:
( y + 3) ( y + 3) = (y - 2)
Now use the standard distributive property to combine like terms:
y2 + 6y + 9 = y – 2
Use addition equality property:
y2 + 6y + 9 – y + 2 = 0
y2 + 5y + 11 = 0
Now the equation is in standard form so we can implement the quadratic formula, first grab the values of a, b and c from the equation.
a = 1, b = 5 and c = 11.
Put this value in the formula
; by using the ‘i’ definition roots can be easily determined.This is all about Quadratic equations and functions. In this article we have explored all the possible ways of solving any quadratic equation. These four ways of evaluating quadratic equation covers the topic of quadratic function in grade XII math. Students of this class most of the times fixed in troubles because there are several ways in which a similar problem can be asked and every time there is a bit of changes in the solution process, so it is hard to memorize all the ways and forms of quadratic equation. This task only can be accomplished by doing practice as much they can and to help students in it Online math tutor websites do their role by giving immediate assistance and various worksheets to solve. If any kind of doubt stays in student's mind related to any topic then the whole fundamentals gets messed up, so it must be required that student asks his queries on regular basis and this will happen only when student does not hesitate in front of tutor which most often happen. To troubleshoot this problem, Online services uses online Chatting option via text or video, to build a better conversation between students and tutors, and because most of us are friendly with Internet platform, the access of this service gets easier.
In grade XII math content Algebra is covering most of the problem region so it gets more essential to learn its each and every problem with all the possible ways of query presentation. For having the detailed descriptions of other topics of Grade XII math syllabus, keep following the successive articles that will surely add a wide range of knowledge.
In upcoming posts we will discuss about Exponents in on Grade XII and Methods of data representation. Visit our website for information on Maharashtra state board
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