Factor completely
(x-2/3)^2-16(x-2/3)+63
Substitution: z = x - 2 / 3
( x - 2 / 3 )² - 16 ( x - 2 / 3 ) + 63 = z² - 16 z + 63
Now you must solve equation:
z² - 16 z + 63 = 0
z1/2 = [ - b ± √ ( b² - 4 a c ) ] / 2 a
In this case:
a = 1 , b = - 16 , c = 63
z1/2 = [ - ( - 16 ) ± √ ( ( - 16 )² - 4 ∙ 1 ∙ 63 ) ] / 2 ∙ 1 =
[ 16 ± √ ( 256 - 252 ) ] / 2 =
( 16 ± √ 4 ) / 2 =
( 16 ± 2 ) / 2 =
8 ± 1
z1 = 8 + 1 = 9
z2 = 8 - 1 = 7
z² - 16 z + 63 = ( z - z1 ) ( z - z2 ) =
z² - 16 z + 63 = ( z - 9 ) ( z - 7 ) =
z² - 16 z + 63 = ( z - 7 ) ( z - 9 )
Return to the initial variables:
z = x - 2 / 3
( x - 2 / 3 )² - 16 ( x - 2 / 3 ) + 63 =
z² - 16 z + 63 =
( z - 7 ) ( z - 9 ) =
( x - 2 / 3 - 7 ) ( x - 2 / 3 - 9 ) =
( x - 2 / 3 - 21 / 3 ) ( x - 2 / 3 - 27 / 3 ) =
( x - 23 / 3 ) ( x - 29 / 3 ) =
( 3 x / 3 - 23 / 3 ) ( 3 x / 3 - 29 / 3 ) =
( 1 / 3 ) ∙ ( 3 x - 23 ) ∙ ( 1 / 3 ) ∙ ( 3 x - 29 ) =
( 1 / 3 ) ∙ ( 1 / 3 ) ∙ ( 3 x - 23 ) ∙ ( 3 x - 29 ) =
( 1 / 9 ) ∙ ( 3 x - 23 ) ∙ ( 3 x - 29 )
( x - 2 / 3 )² - 16 ( x - 2 / 3 ) + 63 = ( 1 / 9 ) ( 3 x - 23 ) ( 3 x - 29 )
To factor the expression completely, use the factoring method or quadratic formula. However, before proceeding, let's simplify the expression first:
Let's expand the square of the first term:
(x - 2/3)^2 = (x - 2/3)(x - 2/3) = x^2 - (2/3)x - (2/3)x + (2/3)(2/3) = x^2 - (4/3)x + 4/9
Now, let's focus on the original expression:
(x - 2/3)^2 - 16(x - 2/3) + 63
= (x^2 - (4/3)x + 4/9) - 16(x - 2/3) + 63
= x^2 - (4/3)x + 4/9 - 16x + 32/3 + 63
= x^2 - (52/3)x + 207/3
Now, let's factor the expression:
x^2 - (52/3)x + 207/3
To factor, we need to find two numbers whose sum is equal to -52/3 and whose product is equal to 207/3. Let's express the middle term (-52/3) as the sum of two terms, which will help us in factorization.
The factors of 207/3 can be found by listing all the pairs of numbers whose product equals 207/3. After finding such a pair, we need to check if their sum equals -52/3.
The factor pairs for 207/3 are:
1/3, 207/3
3/3, 69/3
69/3, 3/3
207/3, 1/3
None of these pairs sum up to -52/3, so we conclude that the expression cannot be factored further using rational numbers.
Therefore, the factored form of the expression would be:
x^2 - (52/3)x + 207/3