Solve The equation for x

x = 6 or x = 11

Step-by-step explanation:

Solve for x:

sqrt(x - 2) = 5 - sqrt(15 - x)

Raise both sides to the power of two:

x - 2 = (5 - sqrt(15 - x))^2

Subtract (5 - sqrt(15 - x))^2 from both sides:

-2 - (5 - sqrt(15 - x))^2 + x = 0

-2 - (5 - sqrt(15 - x))^2 + x = -42 + 10 sqrt(15 - x) + 2 x:

-42 + 10 sqrt(15 - x) + 2 x = 0

Simplify and substitute y = sqrt(15 - x).

-42 + 10 sqrt(15 - x) + 2 x = -12 + 10 sqrt(15 - x) - 2 (sqrt(15 - x))^2

= -2 y^2 + 10 y - 12:

-2 y^2 + 10 y - 12 = 0

The left hand side factors into a product with three terms:

-2 (y - 3) (y - 2) = 0

Divide both sides by -2:

(y - 3) (y - 2) = 0

Split into two equations:

y - 3 = 0 or y - 2 = 0

Add 3 to both sides:

y = 3 or y - 2 = 0

Substitute back for y = sqrt(15 - x):

sqrt(15 - x) = 3 or y - 2 = 0

Raise both sides to the power of two:

15 - x = 9 or y - 2 = 0

Subtract 15 from both sides:

-x = -6 or y - 2 = 0

Multiply both sides by -1:

x = 6 or y - 2 = 0

Add 2 to both sides:

x = 6 or y = 2

Substitute back for y = sqrt(15 - x):

x = 6 or sqrt(15 - x) = 2

Raise both sides to the power of two:

x = 6 or 15 - x = 4

Subtract 15 from both sides:

x = 6 or -x = -11

Multiply both sides by -1:

x = 6 or x = 11

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step-by-step explanation: