Question

SOLVE FOR X:
$x - 2 \sqrt{x} \: - 8 = 0$
​

Answer 1

Solution!!

→ x - 2√x - 8 = 0

Move the expression to the right-hand side and change its sign

→ -2√x = 8 - x

Square both sides of the equation

→ (-2√x)² = (8 - x)²

→ (-2)²(√x)² = (8)² + (x)² - 2(8)(x)

→ 4x = 64 + x² - 16x

Move the expression to the left-hand side and

change its sign

→ - x² + 4x + 16x - 64 = 0

Factor out -x and 16 from the expression

→ - x(x - 4) + 16(x - 4) = 0

Factor out -x + 16 from the expression

→ (- x + 16)(x - 4) = 0

Now atleast one of the factors is zero.

→ - x + 16 = 0

→ x - 4 = 0

→ x = 16

→ x = 4

Now, let's check if the value is the solution of the equation.

→ 16 - 2√16 - 8 = 0

→ 4 - 2√4 - 8 = 0

→ 16 - 2(4) - 8 = 0

→ 4 - 2(2) - 8 = 0

→ 16 - 8 - 8 = 0

→ 4 - 4 - 8 = 0

→ 16 - 16 = 0

→ 4 - 12 = 0

→ 0 = 0

→ -8 ≠ 0

Hence, the value of x is 16.

Answer 2

$x = 16$

Topic:

Solving radical equations

Step-by-step explanation:

STEP 1 : Isolate the square root on the left hand side

Original equation

$x-2 \sqrt{x-8}  = 0$

Isolate

$-2 \sqrt{x} = -x+8+0$

Tidy up

$2 \sqrt{x} = x-8$

STEP 2 : Eliminate the radical on the left hand side

Raise both sides to the second power

$(2 \sqrt{x} )2 = (x-8)2$

After squaring

$4x = x2-16x+64$

STEP 3 : Solve the quadratic equation

Rearranged equation

$x2 - 20x + 64 = 0$

This equation has two rational roots:

${x1, x2}={16, 4}$

STEP 4 : Check that the first solution is correct

Original equation, root isolated, after tidy up

$2 \sqrt{x} = x-8$

Plug in 16 for x

$2 \sqrt{(16)} = (16)-8$

Simplify

$2 \sqrt{16} = 8$

Solution checks !!

Solution is:

$x = 16$

STEP 5 : Check that the second solution is correct

Original equation, root isolated, after tidy up

$2 \sqrt{x} = x-8$

Plug in 4 for x

$2 \sqrt{(4)} = (4)-8$

Simplify

$2 \sqrt{4} = -4$

Solution does not check

$4 ≠ -4$

One solution was found :

$x = 16$