Question

solve the equation.

\\x - 2\sqrt{x} - 35 = 0\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\{\quad\\}.
(simplify your answer. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

b. the solution set is the empty set.

Explanation:

🆕 New Concept Discovered: Solving Equations Quadratic in Form
Solving equations that look like quadratics using substitution.

Step 1: Use substitution

We can rewrite the equation by letting a new variable represent the radical term. Let:

$$ u = \sqrt{x} $$

Since \( x = (\sqrt{x})^2 \), we can substitute \( x = u^2 \) into the original equation:

$$ u^2 - 2u - 35 = 0 $$

Step 2: Solve the quadratic equation

Now we solve for \( u \) by factoring the quadratic equation. We look for two numbers that multiply to \(-35\) and add to \(-2\). These numbers are \(-7\) and \(5\):

$$ (u - 7)(u + 5) = 0 $$

This gives two possible values for \( u \):

$$ u = 7 \quad \text{or} \quad u = -5 $$

Step 3: Substitute back and solve for x

Now we substitute \( \sqrt{x} \) back in place of \( u \):

1. For \( u = 7 \):

$$ \sqrt{x} = 7 $$

Squaring both sides gives:

$$ x = 49 $$

1. For \( u = -5 \):

$$ \sqrt{x} = -5 $$

Since the principal square root of a real number, \( \sqrt{x} \), cannot be negative, this equation has no real solution.

Step 4: Verify the solution

Let's check \( x = 49 \) in the original equation:

$$ 49 - 2\sqrt{49} - 35 = 49 - 2(7) - 35 = 49 - 14 - 35 = 0 $$

The solution is valid.

Answer:

A. The solution set is \( \{49\} \).