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Chapter 1: Problem 41

Find the real solutions, if any, of each equation. $$ x^{3 / 2}-3 x^{1 / 2}=0 $$

Short Answer

Expert verified
The real solutions are \( x = 0 \) and \( x = 3 \).

Step by step solution

01

- Substitute Variables

Let us substitute a new variable to simplify the equation. Let \( y = x^{1/2} \). Therefore, \( y^2 = x \). Rewrite the original equation with this substitution: \( y^3 - 3y = 0 \).
02

- Factor the Equation

Factor the equation \( y^3 - 3y = 0 \) by taking out the common factor \( y \): \( y(y^2 - 3) = 0 \).
03

- Solve for y

Set each factor to zero:\( y = 0 \) and \( y^2 - 3 = 0 \). Solve for \( y \) in the second equation:\( y^2 = 3 \)\( y = \pm \sqrt{3} \).
04

- Substitute Back to x

Recall that \( y = x^{1/2} \). So we have three possible values for \( y \):\( y = 0 \), \( y = \sqrt{3} \), and \( y = -\sqrt{3} \).Substitute back to find \( x \):For \( y = 0 \), \( x = 0^2 = 0 \).For \( y = \sqrt{3} \), \( x = (\sqrt{3})^2 = 3 \).For \( y = -\sqrt{3} \), \( x = (-\sqrt{3})^2 = 3 \).
05

- Verify Real Solutions

Check the possible solutions:\( x = 0 \) and \( x = 3 \). Both values are non-negative and thus valid real solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a powerful technique to simplify complicated equations. In this exercise, we start with the equation: $$x^{3/2} - 3x^{1/2} = 0$$. By introducing a new variable, we make the equation easier to handle. Let鈥檚 substitute: $$y = x^{1/2}$$. Given this substitution, the equation transforms to: $$y^3 - 3y = 0$$. This new equation is simpler and more straightforward to solve. The substitution method helps break down complex expressions by converting them to a more familiar form.
Factoring Equations
Factoring is a critical skill for solving polynomial equations. After substitution, we have: $$y^3 - 3y = 0$$. To factor this equation, we look for common factors. In this case, we can factor out a $$y$$: $$y(y^2 - 3) = 0$$. This factored form shows that the equation equals zero when either factor is zero. Identifying and pulling out common factors simplifies the equation and makes it easier to solve for the variable. Effective factoring breaks down the problem into more manageable parts.
Solving Quadratic Equations
Once factored, solving the equation involves finding the values of the variable that satisfy each factor. From the factored equation: $$y(y^2 - 3) = 0$$, we set each factor to zero. First, solve $$y = 0$$. This gives us one solution: $$y = 0$$. Next, solve the quadratic equation: $$y^2 - 3 = 0$$. Rearrange to find: $$y^2 = 3$$, and then take the square root of both sides: $$y = \pm \sqrt{3}$$. These steps yield three possible values for $$y$$: $$0$$, $$\sqrt{3}$$, and $$-\sqrt{3}$$. Quadratic equations often produce multiple solutions because of their parabolic nature.
Verifying Solutions
The final step is to verify the solutions by substituting back into the original variables. Recall that $$y = x^{1/2}$$. For: $$y = 0$$, we get $$x = 0^2 = 0$$. For: $$y = \sqrt{3}$$, we get $$x = (\sqrt{3})^2 = 3$$. For: $$y = -\sqrt{3}$$, we also get $$x = (-\sqrt{3})^2 = 3$$. Thus, we have two unique real solutions for $$x$$: $$0$$ and $$3$$. Verifying solutions involves checking each value against the original equation and ensuring they meet all conditions, confirming their validity as real solutions.

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