Problem 47 - Algebra and Trigonometry

Question

1–54 ? Find all real solutions of the equation. $$ x-\sqrt{x+3}=\frac{x}{2} $$

Step-by-Step Solution

Answer

Only \( x = 6 \) is the valid solution.

Step 1: Understand the Equation

The equation given is \( x - \sqrt{x+3} = \frac{x}{2} \). This is a mix of linear and radical expressions. Our goal is to isolate \( \sqrt{x+3} \) to eliminate the radical.

Step 2: Move Terms to Isolate the Radical

Subtract \( \frac{x}{2} \) from both sides of the equation to get all "x" terms on one side:\[ x - \sqrt{x+3} - \frac{x}{2} = 0 \]This simplifies to:\[ \frac{2x}{2} - \frac{x}{2} - \sqrt{x+3} = 0 \]\[ \frac{x}{2} - \sqrt{x+3} = 0 \]

Step 3: Isolate the Radical Expression

Now, isolate the square root by moving it to the other side of the equation:\[ \frac{x}{2} = \sqrt{x+3} \]

Step 4: Square Both Sides to Remove the Square Root

Square both sides to eliminate the square root:\[ \left( \frac{x}{2} \right)^2 = (\sqrt{x+3})^2 \]This simplifies to:\[ \frac{x^2}{4} = x + 3 \]

Step 5: Clear the Fraction

Multiply every term by 4 to clear the fraction:\[ x^2 = 4(x + 3) \]Now, distribute the 4:\[ x^2 = 4x + 12 \]

Step 6: Arrange into a Quadratic Equation

Rearrange the equation to set it to zero:\[ x^2 - 4x - 12 = 0 \]

Step 7: Use the Quadratic Formula

The quadratic equation \( x^2 - 4x - 12 = 0 \) can be solved using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( a = 1 \), \( b = -4 \), and \( c = -12 \):\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \]\[ x = \frac{4 \pm \sqrt{16 + 48}}{2} \]\[ x = \frac{4 \pm \sqrt{64}}{2} \]\[ x = \frac{4 \pm 8}{2} \]

Step 8: Solve for x

Solving the quadratic formula gives two potential solutions for \( x \):\[ x = \frac{4 + 8}{2} = 6 \]\[ x = \frac{4 - 8}{2} = -2 \]

Step 9: Check for Valid Solutions

Substitute \( x = 6 \) back into the original equation:\[ 6 - \sqrt{6+3} = \frac{6}{2} \]\[ 6 - 3 = 3 \]This holds true. Now check \( x = -2 \):\[ -2 - \sqrt{-2+3} = \frac{-2}{2} \]\[ -2 - 1 eq -1 \]This solution is not valid as it doesn't satisfy the original equation.

Key Concepts

Quadratic Equation

When solving equations with variables squared, you're often dealing with a quadratic equation. In our problem, we ended up with the quadratic equation \[ x^2 - 4x - 12 = 0 \]. A quadratic equation is a second-degree polynomial equation where the highest degree of the variable is 2. These equations can have 0, 1, or 2 real solutions depending on their discriminant.

• The general form is: \( ax^2 + bx + c = 0 \)
• Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).

Solving quadratic equations can be done through factoring, completing the square, or using the quadratic formula. It’s crucial to correctly set the equation to zero before applying any of these methods.

Linear and Radical Expressions

In the original equation \( x - \sqrt{x+3} = \frac{x}{2} \), we see elements of both linear and radical expressions. A linear expression involves variables to the first power and no radicals. On the other hand, a radical expression includes a square root or another root.

• A linear expression example: \( x - \frac{x}{2} \)
• A radical expression example: \( \sqrt{x+3} \)

When you encounter an equation with both types, your goal is to isolate and simplify them. Here, we focused on isolating the square root to handle the radical expression separately, which often simplifies the solving process.

Square Root Isolation

Square root isolation is a technique to handle equations with radical components effectively. In our exercise, we modified the equation to end up with \[ \frac{x}{2} = \sqrt{x+3} \]. This allowed us to focus purely on simplifying and solving the radical part.
Here's why isolating the square root is helpful:

• It makes it easier to eliminate the square root by squaring both sides of the equation.
• The process simplifies the equation to a form that's more familiar, like a quadratic form.

After isolating the square root, you want to be cautious when squaring both sides, as this can sometimes introduce extraneous solutions, which need checking later.

Quadratic Formula Application

The quadratic formula is a powerful tool for solving quadratic equations that cannot be factored easily. In our problem, after arranging our equation \[ x^2 - 4x - 12 = 0 \], we used the quadratic formula. This formula states:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substitute the coefficients \( a, b, \) and \( c \) from your quadratic equation into the formula. In our example, this was:

• \( a = 1 \)
• \( b = -4 \)
• \( c = -12 \)

The formula then yielded two potential solutions, of which only those satisfying the original equation were accepted. It's essential to calculate carefully and verify solutions in real contexts, to ensure they meet problem requirements.