solve sqrt(x) + 1 = y for x — step-by-step solution

1) Let's solve in

ℝ

\sqrt{x} + 1 = y

Isolate the square root:

\sqrt{x} = y - 1

. Since the square root is nonnegative, square both sides, provided that

y - 1 \geq 0

x = {(y - 1)}^{2}

ⓘ We subtract

0

from both sides■Raising a sum to an integer power is multiplying it by itself:

{(y - 1)}^{2}

(y - 1) \cdot (y - 1)

x = (y - 1) \cdot (y - 1)

ⓘ Expand by multiplying each term inside the parentheses:

(y - 1) \cdot (y - 1)

y \cdot y

+ y \cdot (-1)

- 1 \cdot y

- 1 \cdot (-1)

x = y \cdot y

+ y \cdot (-1)

- 1 \cdot y

- 1 \cdot (-1)

ⓘ Multiplying powers with the same base adds the exponents:

y \cdot y

y^{2}

■Multiplying by -1 gives the opposite:

y \cdot (-1)

- y

■Multiplying by 1 changes nothing:

1 \cdot y

y

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-1 \cdot (-1)

1

x = y^{2}

- y

- y

+ 1

ⓘ Like terms are grouped by adding their coefficients:

- y - y

-2 y

Candidates, pending the domain check:

x = y^{2} - 2 y + 1

2) Solving for y:

y - 1 \geq 0

ⓘ We add

1

to both sides

Candidates, pending the domain check:

y \geq 1

For the expression

\sqrt{x}

to exist, you must check:

x \geq 0

Domain constraints:

x \geq 0

▶Solution:

{x = y^{2} - 2 y + 1, y \geq 1}