Find all solutions to the equation,
x-8(sqrt x)+16=0. Is x=16 the only solution?
To find all solutions to the equation, we need to solve it step by step.
Step 1: Start with the given equation:
x - 8√x + 16 = 0
Step 2: Rearrange the equation to isolate the square root term:
8√x = x + 16
Step 3: Square both sides of the equation to eliminate the square root:
(8√x)^2 = (x + 16)^2
64x = x^2 + 32x + 256
Step 4: Rearrange the equation to set it equal to zero:
x^2 - 32x + 256 - 64x = 0
x^2 - 96x + 256 = 0
Step 5: Factor the quadratic equation:
(x - 16)(x - 16) = 0
(x - 16)^2 = 0
Step 6: Solve for x:
x - 16 = 0
x = 16
Step 7: Check for extraneous solutions (solutions that do not satisfy the original equation):
Insert x = 16 back into the original equation:
16 - 8√16 + 16 = 0
16 - 8(4) + 16 = 0
16 - 32 + 16 = 0
0 = 0
The solution x = 16 satisfies the original equation and is the only solution. There are no extraneous solutions. Therefore, x = 16 is the only solution to the equation x - 8√x + 16 = 0.
To find all solutions to the equation x - 8√x + 16 = 0, we need to solve for x.
Let's solve it step by step:
1. Start by isolating the term containing the square root. Move the constant term to the other side of the equation:
x - 8√x = -16
2. To eliminate the square root, we can square both sides of the equation. However, remember that when we square both sides of an equation, we might introduce extraneous solutions:
(x - 8√x)^2 = (-16)^2
x^2 - 16x + 64 = 256
3. Simplify the equation:
x^2 - 16x - 192 = 0
4. Now we have a quadratic equation. To solve it, we can factor it or use the quadratic formula.
We'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -16, c = -192.
5. Substitute the values into the quadratic formula:
x = (-(-16) ± √((-16)^2 - 4 * 1 * -192)) / (2 * 1)
x = (16 ± √(256 + 768)) / 2
x = (16 ± √1024) / 2
x = (16 ± 32) / 2
6. Simplify:
x = (16 + 32) / 2 OR x = (16 - 32) / 2
x = 48 / 2 OR x = -16 / 2
x = 24 OR x = -8
Thus, the equation x - 8√x + 16 = 0 has two solutions, x = 24 and x = -8. Therefore, x = 16 is not the only solution.
8 sqrt x = x+16
64 x = x^2 + 32 x + 256
x^2 -32 x + 256 = 0
(x- 16)^2 = 0
x = 16 is the only root I see