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.
 

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

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.

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.

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.

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.

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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