Solving Equation: x = sqrt(3x + x^2 - 3sqrt(3x + x^2))

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In summary, to solve the equation x = sqrt(3x + x^2 - 3sqrt(3x + x^2)), we can first rewrite it and then use the quadratic formula to find its solutions. This equation can also be solved using graphing or numerical methods. It has various real-life applications and is restricted to values of x ≥ -1 and x ≤ 3.
  • #1
mathdad
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Find all real solutions of the equation.

$x = \sqrt{3x + x^2 - 3\sqrt{3x + x^2}}$

Must I square each side twice to start?

$(x)^2 = [\sqrt{3x + x^2 - 3\sqrt{3x + x^2}}]^2$

$x^2 = 3x + x^2 - 3\sqrt{3x + x^2}$

$x^2 - x^2 - 3x = -3\sqrt{3x + x^2}$

$-3x = -3\sqrt{3x + x^2}$

$x = \sqrt{3x + x^2}$

$(x)^2 = [\sqrt{3x + x^2}]^2$

$x^2 = 3x + x^2$

$x^2 - x^2 = 3x$

$0 = 3x$

$0/3 = x$

$0 = x$

Is this right?
 
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  • #2
Yes, it's correct.
 
  • #3
Good to know that I am right.
 

Related to Solving Equation: x = sqrt(3x + x^2 - 3sqrt(3x + x^2))

1. How do I solve the equation x = sqrt(3x + x^2 - 3sqrt(3x + x^2))?

To solve this equation, we can first rewrite it as x = √(3x + x² - 3√(3x + x²)). Then, we can square both sides of the equation to get x² = 3x + x² - 3√(3x + x²). Simplifying, we get 3√(3x + x²) = 3x. Dividing both sides by 3, we get √(3x + x²) = x. Now, we can square both sides again to eliminate the square root and solve for x. We get 3x + x² = x². Simplifying, we get x = 0. Therefore, the solution to the equation is x = 0.

2. Is there a simpler way to solve this equation?

Yes, there is a simpler way to solve this equation. We can use the quadratic formula to solve for x. First, we can rewrite the equation as x = √(3x + x² - 3√(3x + x²)). Squaring both sides, we get x² = 3x + x² - 3√(3x + x²). Simplifying, we get 3√(3x + x²) = 3x. Dividing both sides by 3, we get √(3x + x²) = x. Now, we can square both sides again to eliminate the square root. We get 3x + x² = x². Simplifying, we get x² - 3x = 0. This is now in the form ax² + bx + c = 0, where a = 1, b = -3, and c = 0. Plugging these values into the quadratic formula, we get x = 0 or x = 3. Therefore, the solutions to the equation are x = 0 or x = 3.

3. Can this equation be solved using any other methods?

Yes, this equation can also be solved using graphing or numerical approximation methods. By graphing the equation, we can find the x-intercepts, which represent the solutions to the equation. We can also use numerical methods, such as Newton's method or the bisection method, to approximate the solutions to the equation.

4. What real-life applications does this equation have?

This equation can be used to model various real-life situations, such as calculating the time it takes for an object to reach a certain height when thrown with a specific initial velocity, or determining the amount of time it takes for a substance to decay based on its initial concentration and decay rate.

5. Are there any restrictions on the values of x for this equation?

Yes, there are restrictions on the values of x for this equation. Since there is a square root in the equation, the value inside the square root must be greater than or equal to 0. Therefore, the values of x that satisfy this condition are x ≥ -1. Additionally, when using the quadratic formula, we must also consider the restriction for the discriminant (b² - 4ac) to be greater than or equal to 0. This gives us the additional restriction that x ≤ 3. Therefore, the values of x that satisfy both of these conditions are -1 ≤ x ≤ 3.

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