91Ó°ÊÓ

One crucial step when solving equations involving square roots is isolating the square root term. This straightforward yet essential process ensures we can effectively apply further operations to solve the equations.
To begin isolating a square root, rearrange the equation so that the term containing the square root stands alone on one side. In our initial equation, we start with \( x - \sqrt{19 - 2x} - 2 = 0 \).
By moving all other terms to the other side, we reach \( \sqrt{19 - 2x} = x - 2 \).
This isolation makes it practical to eliminate the square root through squaring both sides, a crucial step to advance toward finding a solution. Isolating the square root simplifies further computation and lessens potential errors.

Extraneous solutions often arise when solving equations involving square roots or rational expressions. These are solutions that appear during the algebraic manipulation process but do not satisfy the original equation.
To identify such solutions, it is essential to substitute the potential solutions back into the original equation. In our example, we obtained two potential solutions, \( x = 5 \) and \( x = -3 \).
Verification shows that \( x = 5 \) satisfies the equation, while \( x = -3 \) does not. Substituting \( x = -3 \) back into the equation results in a false statement since:

• \(-3 - \sqrt{19 - 2(-3)} - 2 = -10\), not zero.

This means \( x = -3 \) is an extraneous solution. Checking solutions not only verifies correctness but also eliminates these spurious results.

The quadratic formula is a method for finding the roots of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \).
It is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. In our example, after squaring both sides and simplifying, we derived the quadratic form \( x^2 - 2x - 15 = 0 \).
Here, \( a = 1 \), \( b = -2 \), and \( c = -15 \).
The quadratic formula provides reliable solutions for any quadratic equation, especially when factoring is not an option or seems overly complicated. The key benefit of using this formula is its ability to handle complex numbers when necessary, though this wasn’t needed for this particular problem. By plugging the given values into the formula, we calculate the potential roots, enhancing our solving toolkit.

The discriminant, denoted by \( b^2 - 4ac \) within the quadratic formula, reveals the nature and number of solutions for a quadratic equation.
In our problem, calculating the discriminant with \( a = 1 \), \( b = -2 \), and \( c = -15 \) gives: \( (-2)^2 - 4 \cdot 1 \cdot (-15) = 64 \).
This positive and perfect square (\( 64 = 8^2 \)) indicates two distinct real solutions exist.
Understanding the discriminant's role is critical as:

• A positive perfect square means two real, rational roots.
• A positive non-perfect square means two real, irrational roots.
• Zero means one real rational root (a repeated or double root).
• A negative value means two complex roots.

The discriminant's analysis simplifies understanding solutions without solving the equation completely, proving essential in determining the equation's dynamics.