(x-1)^2=7 solve quadratic equations by factoring..........?

Answer 1

take squre root of both sides
x-1=+ or - sqrt(7)

then
the first root
x1=1-sqrt(7)

the second root
x2=1+sqrt(7)

By factoring method:

take 7 to left
(x-1)^2-(sqrt(7))^2=0
factor like a^2-b^2=(a-b)(a+b)

[x-1-sqrt(7)] * [x-1+sqrt(7)]=0
0*0=0
x=1+sqrt(7)
or
x=1-sqrt(7)

Answer 2

This particular quadratic equation cannot be solved by _factoring_ in the usual sense.
To factor we set one side equal to zero:
(x-1)^2 - 7 = 0
x^2 - 2x -6 = 0
and then try to factor the left-hand side.

It turns out that this quadratic: x^2 - 2x + 6 is prime, so factoring will not work.

The only hope to factor is to consider 7 as sqrt( 7 )^2
Then we have
(x-1)^2 - sqrt(7)^2 = 0.
The left hand side of this equation is a difference of two squares, so it factors as

( (x-1) + sqrt(7) ) * ( (x-1) - sqrt(7) ) = 0
Thus
( x-1 ) + sqrt( 7 ) = 0 or ( x-1 ) - sqrt( 7 ) = 0
( x-1 ) = - sqrt( 7 ) or ( x-1 ) = sqrt( 7 )
x = 1 - sqrt( 7 ) or x = 1 + sqrt( 7 )
But this was not the "standard" factoring as polynomials with integer coefficients.

After the equation is in the form
(x-1)^2 = 7
the quickest way to solve it is to take square roots of both sides.
We then get
(x-1) = +/- sqrt( 7 )
x = 1 +/- sqrt( 7 )

By the way, as these solutions are irrational we automatically know that we could not factor the earlier quadratic polynomial.

Answer 3

First, FOIL out the (x-1)^2 and combine like terms. Then work out the equation so that one side is equal to 0 by combining like terms and you get:
x^2 - 2x - 6 = 0
Then, use the quadratic formula:
x = (2 ± sqrt(2*2 - 4(-6)1))/2
x = (2 ± sqrt(4 + 24))/2
x = (2 ± sqrt(28))/2
x = (2 ± 2*sqrt(7))/2
x = 1 ± sqrt(7)

Answer 4

(x - 1)^2 - 7 = 0
x^2 - 2x + 1 - 7 = 0
x^2 - 2x - 6 = 0

Can't factor, solve using quadratic formula:

a = 1; b = -2; c = -6

{-(-2) +/- sqrt [(-2)^2 - 4(1)(-6)]} / 2(1) =
[2 +/- sqrt (4 + 24)] / 2 = (2 +/- sqrt 28) / 2 = 1 +/- sqrt 7

Answer 5

You can't factor it anymore. Try expanding.

x²-2x+1 = 7...
x²-2x-6 = 0...
x = [2 +- sqrt(4 + 24)] / 2...
x = 1 +- sqrt(7)...

Now, to take the LRHS approach...

x-1 = sqrt(7)
x = sqrt(7) + 1

so the answer is x = sqrt(7) + 1.

Answer 6

x=sqrt(7)+1

Answer 7

(x - 1)^2 = 7
(x - 1)(x - 1) = 7
x^2 - x - x + 1 = 7
x^2 - 2x + 1 = 7
x^2 - 2x - 6 = 0

This problem cannot be factored

Answer 8

(x-1)^2=7 sqrt both sides

x-1=+sqrt(7) or x-1=-sqrt(7)
x=+sqrt(7)+1 or x=-sqrt(7)+1