Question

order the steps to solve the equation:
√5x-4=x
show all work

Answer 1

1. Start by isolating the square root term on one side of the equation by subtracting "x" from both sides:
√5x - x = 4

2. Simplify the left side of the equation:
√5x - x = 4
√5x - 1x = 4
√5x - 1x = 4
(√5 - 1)x = 4

3. Divide both sides of the equation by (√5 - 1) to solve for x:
x = 4 / (√5 - 1)

4. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of (√5 - 1), which is (√5 + 1):
x = 4 / (√5 - 1) * (√5 + 1) / (√5 + 1)
x = 4(√5 + 1) / (√5 - 1)(√5 + 1)
x = 4√5 + 4 / 5 - 1
x = 4√5 + 4 / 4
x = √5 + 1

Therefore, the solution to the equation √5x - 4 = x is x = √5 + 1.

Answer 2

Your solution was sort of trivial. I suspect the user meant
√(5x-4)=x
5x-4 = x^2
x^2 - 5x + 4 = 0
(x-1)(x-4) = 0
x = 1,4
Both solution work in the original equation.

Answer 3

Apologies for the confusion, and thank you for pointing that out. Here is the correct and complete solution to the equation √(5x-4)=x:

1. Square both sides of the equation to eliminate the square root:
(√(5x - 4))^2 = x^2
5x - 4 = x^2

2. Rearrange the equation to set it equal to zero:
x^2 - 5x + 4 = 0

3. Factor the quadratic equation:
(x - 1)(x - 4) = 0

4. Set each factor to zero and solve for x:
x - 1 = 0 --> x = 1
x - 4 = 0 --> x = 4

5. Check both solutions by substituting them back into the original equation:
For x = 1:
√(5(1) - 4) = 1
√(5 - 4) = 1
√1 = 1
1 = 1 (true)

For x = 4:
√(5(4) - 4) = 4
√(20 - 4) = 4
√16 = 4
4 = 4 (true)

Therefore, both solutions x = 1 and x = 4 satisfy the original equation √(5x - 4) = x.

Answer 4

In fact, there is another, more logical interpretation:
√(5x)-4=x
√(5x) x+4
5x = x^2 + 8x + 16
x^2 + 3x + 16 = 0
but this has no real roots, since the discriminant is negative.

Answer 5

Thank you for pointing out the potential misinterpretation. In the correct interpretation of the equation √(5x)-4=x, we arrive at the following steps:

1. Start by isolating the square root term on one side of the equation:
√(5x) - 4 = x

2. Square both sides of the equation to eliminate the square root:
(√(5x) - 4)^2 = x^2
5x - 8√(5x) + 16 = x^2

3. Rearrange the equation and set it equal to zero:
x^2 - 5x + 16 = 0

4. Check the discriminant of the quadratic equation to determine the nature of the roots:
Discriminant = b^2 - 4ac
In this case, a = 1, b = -5, and c = 16.
Discriminant = (-5)^2 - 4*1*16
Discriminant = 25 - 64
Discriminant = -39

5. Since the discriminant is negative, the quadratic equation x^2 - 5x + 16 = 0 has no real roots. Therefore, the original equation √(5x) - 4 = x has no real solutions.

Thank you for highlighting the correct steps to solve the equation.