Question

How do you solve `(x - 1)^2 = 9`?

Answer 1

x1= 4 and x2=-2

Explanation:

x^2 - 2x*1 +1^2 = 9
x^2-2x+1=9

since this is a quadratic equation it has to be equal to 0, so:

x^2-2x+1-9=0
x^2-2x-8=0

we can solve this in two ways: the first way is to find the factors of x^2-2x-8 and the second one is to solve it by using the quadratic formula.

Let me factorize it.
(x-4)(x+2)=0

In order for this to be equal zero:
x-4=0 => x=4
and
x+2=0 => x=-2

The quadratic equation has always two roots or two solutions. Consequently, x1= 4 and x2=-2.

Answer 2

`x=4,-2`

Explanation:

`color(blue)((x-1)^2=9`

Use the golden rule of Algebra: What you do on one side must be done on the other side also

Take the square root of both sides (`sqrt`)

`rarrsqrt((x-1)^2=sqrt9`

Use: `color(brown)(sqrt(x^2)=x`

`rarrx-1=sqrt9`

But, `color(brown)(sqrt9=+-sqrt9=(3,-3)`

Solve for both

`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

(`3`)

`rarrx-1=3`

Add `1` both sides

`rarrxcancel(-1color(purple)(+1))=3color(purple)(+1)`

`rarrx=3+1`

`color(green)(rArrx=4`

`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

(`-3`)

`rarrx-1=-3`

Add `1` both sides

`rarrxcancel(-1color(purple)(+1))=-3color(purple)(+1`

`color(green)(rArrx=-2`

`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

`color(blue)( ul bar |x=(4,-2)|`