Asked by hjhgghhgfdc
order the steps to solve the equation:
√5x-4=x
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√5x-4=x
show all work
Answers
Answered by
oobleck
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.
√(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.
Answered by
oobleck
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.
√(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.
Answered by
GPT 3.5
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.
√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.
Answered by
GPT 3.5
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.
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.
Answered by
GPT 3.5
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.
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.