Completing the Squares is a technique used in algebra to convert quadratic equations into a form which is sum of square of a linear equation and a constant as:
Ax^2 + bx + c = a(linear equation)2 + C
First let us see what is the expansion of (x+y)2 and (x-y)2.
(x+y)^2 = x^2 + y2 + 2xy
(x-y)^2 = x^2 + y2 - 2xy
For example (x+3)2 = x^2 + 32 +2.3x = x^2+6x+9.
(x-3)2 = x^2 + 32 - 2.3x = x^2-6x+9.
Let us see method of completing squares:
Suppose you are given a quadratic equation ax^2 + bx + c = 0 which is not a perfect square. Then follow the following steps:
1) Make the coefficient of x equal to 1 by diving the equation by its coefficient which is ‘a’ in this case:
x^2 + (b/a)x + c/a =0
2) Write the coefficient of c as multiple of 2 as:
x^2 + 2()x + (c/a) = 0…………..(1)
3) Now write the perfect square equation for which first term is x and second is coefficient of x divided by 2 which is b/2a here:
(x+(b/2a)2 )
This on expansion gives: x^2++2()x = x^2 + x + ………..(2)
Note that in equation (1) and (2), first and second terms are equal and only difference is in constant term.
4) Now write the original equation as (x + )2 + ( - ) = 0
Having problem with solve algebra problems online keep reading my upcoming posts, i will try to help you.
Let us take an example:
x^2 + 10x + 28
Here coefficient of x is already 1 so we will come to step (2) directly. Make the coefficient of x as factor of 2. So, equation can be written as:
x^2 + 2(5)x + 28 ………….(1)
Coming to step 3 we write a perfect square equation as:
(X+5)2 which gives x^2 + 52 + 2.5x on expansion. So we have x^2 + 10x + 25 ………(2) as the expansion now.
Comparing equation 1 and equation 2 we get a difference of 3 in constant term as 28-25 = 3. So, x^2 + 10x + 25 + 3 = equation (1) = x^2 + 2(5)x + 28
Also writing the LHS of above equation in square form we get: (x+5)2 + 3 = x^2 + 10x + 28.
Completing the Square Practice is easy to do as we see in above steps.
You can also Factor by Completing the Square as in this example: x^2 – 4x – 8 = 0. as done in above Math Completing the Square of this equation we get: (x-2)2=12. X-2=(12)1/2 which gives x= 2+121/2.
Ax^2 + bx + c = a(linear equation)2 + C
First let us see what is the expansion of (x+y)2 and (x-y)2.
(x+y)^2 = x^2 + y2 + 2xy
(x-y)^2 = x^2 + y2 - 2xy
For example (x+3)2 = x^2 + 32 +2.3x = x^2+6x+9.
(x-3)2 = x^2 + 32 - 2.3x = x^2-6x+9.
Let us see method of completing squares:
Suppose you are given a quadratic equation ax^2 + bx + c = 0 which is not a perfect square. Then follow the following steps:
1) Make the coefficient of x equal to 1 by diving the equation by its coefficient which is ‘a’ in this case:
x^2 + (b/a)x + c/a =0
2) Write the coefficient of c as multiple of 2 as:
x^2 + 2()x + (c/a) = 0…………..(1)
3) Now write the perfect square equation for which first term is x and second is coefficient of x divided by 2 which is b/2a here:
(x+(b/2a)2 )
This on expansion gives: x^2++2()x = x^2 + x + ………..(2)
Note that in equation (1) and (2), first and second terms are equal and only difference is in constant term.
4) Now write the original equation as (x + )2 + ( - ) = 0
Having problem with solve algebra problems online keep reading my upcoming posts, i will try to help you.
Let us take an example:
x^2 + 10x + 28
Here coefficient of x is already 1 so we will come to step (2) directly. Make the coefficient of x as factor of 2. So, equation can be written as:
x^2 + 2(5)x + 28 ………….(1)
Coming to step 3 we write a perfect square equation as:
(X+5)2 which gives x^2 + 52 + 2.5x on expansion. So we have x^2 + 10x + 25 ………(2) as the expansion now.
Comparing equation 1 and equation 2 we get a difference of 3 in constant term as 28-25 = 3. So, x^2 + 10x + 25 + 3 = equation (1) = x^2 + 2(5)x + 28
Also writing the LHS of above equation in square form we get: (x+5)2 + 3 = x^2 + 10x + 28.
Completing the Square Practice is easy to do as we see in above steps.
You can also Factor by Completing the Square as in this example: x^2 – 4x – 8 = 0. as done in above Math Completing the Square of this equation we get: (x-2)2=12. X-2=(12)1/2 which gives x= 2+121/2.
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