Problem 57 - Algebra and Trigonometry

Question

\(5-60\) Find all real solutions of the equation. $$ x^{2} \sqrt{x+3}=(x+3)^{3 / 2} $$

Step-by-Step Solution

Answer

The real solutions are \(x = \frac{1 + \sqrt{13}}{2}\) and \(x = \frac{1 - \sqrt{13}}{2}\) with \(x \geq -3\).

Step 1: Understand the Given Equation

We are given the equation \(x^2 \sqrt{x+3} = (x+3)^{3/2}\). Notice that the right side \((x+3)^{3/2}\) can be rewritten as \(((x+3)^{1/2})^3\). This might help in simplifying the equation.

Step 2: Simplify Both Sides

First, let's rewrite the left side of the equation. We have \(x^2 \sqrt{x+3}\), which is also \(x^2 (x+3)^{1/2}\). The right side remains \((x+3)^{3/2}\).

Step 3: Equate the Simplified Terms

Since both sides involve \(\sqrt{x+3}\), notice that \((x+3)^{3/2} = (x+3)(x+3)^{1/2}\). Now, divide both sides by \((x+3)^{1/2}\) given it's not zero: \((x^2 = x+3)\).

Step 4: Solve the Quadratic Equation

We now solve the equation \(x^2 = x+3\). Rearrange it to form a standard quadratic equation: \(x^2 - x - 3 = 0\). Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-1\), and \(c=-3\).

Step 5: Apply the Quadratic Formula

Substitute \(a=1, b=-1, c=-3\) into the quadratic formula: \[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1}\] which simplifies to \[x = \frac{1 \pm \sqrt{1 + 12}}{2}\]. Calculate further: \[x = \frac{1 \pm \sqrt{13}}{2}\].

Step 6: Determine the Solution Set

The solutions from the quadratic formula are \(x = \frac{1 + \sqrt{13}}{2}\) and \(x = \frac{1 - \sqrt{13}}{2}\). However, we must check if these satisfy the original condition that \(x+3 \geq 0\) by determining if \(x \geq -3\).

Key Concepts

Real Solutions

When dealing with equations, especially quadratic ones, we often talk about real solutions. A real solution is a value for the unknown variable that satisfies the equation, using real numbers. For the equation in this problem, you first simplify it to make it easier to find these values. The simplification process often helps in determining if potential solutions are viable.

• This involves treating both sides of the equation equally, ensuring no mathematical properties are ignored.
• Remember, an important part of identifying real solutions is ensuring the expressions within square roots, denominators, or logarithms maintain their mathematical values within their allowable domains.
• In this problem, once simplified, the equation is further solved using standard techniques to verify real solutions.

Calculating solutions doesn't guarantee they are feasible within the context of the problem, so always verify if they satisfy the conditions set by the original equation.

Square Roots

Square roots are fundamental to this problem. Understanding them allows you to simplify and adjust equations like the one in the exercise. A square root of a number is a value that, when multiplied by itself, gives the original number. This is key when working with terms involving square roots like \(\sqrt{x+3}\).
Square roots introduce constraints in equations because the expression inside the square root must be non-negative (in real numbers). This means

• For square roots, any expression under the radical (like \(x+3\) in the equation) needs to be checked to ensure it doesn't become negative.
• This introduces an additional step in solving, which involves determining and checking the domain of the solution.
• After simplifying using square roots, solutions should be restricted only to numbers that maintain the validity of the original equation.

Through understanding square roots, including their application and constraints, accurate solutions are more attainable and verify real solutions.

Quadratic Formula

The quadratic formula is a powerful method for finding the real solutions to a quadratic equation. It uses the coefficients of a quadratic equation \(ax^2 + bx + c = 0\) to find unknown value(s) of \(x\). Here's what you need to know:

• The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• It's crucial because it provides a direct solution for any quadratic equation, avoiding trial and error.
• Each part of the formula plays a role – the \(-b\) and \(\pm\) considers both potential solutions inherent in quadratics.
• The expression under the square root sign, \(b^2 - 4ac\), known as the discriminant, tells us about the nature of the roots.

In this problem's equation, the quadratic formula elegantly finds values for \(x\), showing two possible solutions. It appends a critical step by checking mathematical consistency with initial conditions. Thus, ensuring validity of any solution produced using quadratic equations.