Solving the Equation: Steps and Insights for sqrt{5-x} 5-x^2

In this article, we delve into the solution of the equation sqrt{5-x} 5-x^2. We'll explore the algebraic manipulation, simplification techniques, and insights that lead to the correct solutions. This process involves squaring both sides, factoring, and handling nested square roots.

Step 1: Initial Equation

The given equation is:

5-x5-x2

Step 2: Squaring Both Sides

First, we square both sides of the equation to eliminate the square root:

5-x2(5-x2)2

This simplifies to:

5-x25-1 x4

Which further simplifies to:

x4-1 200

Step 3: Substitution and Factoring

Let's rewrite the equation using substitution:

5-xt2;x5-t2

Substitute this into the original equation:

t25-5-t2

This gives:

t4-10t 200

Now, we can factor this equation or solve for (t) using the quadratic formula. From the quadratic equation ((t^2 - 5)t - 4 0), we get:

t-1pm212,t1pm172

Step 4: Solving for (x)

From the substitution (x 5 - t^2), we now solve for (x). We get:

x-1pm212,x1pm172

We need to check which of these values satisfy the original equation. Let's verify:

Value 1:

x-1 212

Value 2:

x1-172

Final Solutions

After simplification and verification, the valid solutions are:

x-1 212

Note: Approximations for these values are:

x ≈ 1.7913

x ≈ 1.5616

These solutions are derived by considering the valid roots that satisfy the original equation.