Problem 45 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{5 x+10}=-5 $$

Step-by-Step Solution

Answer

The given equation does not have any real solutions because the square root of a number cannot be negative. Therefore, any potential solutions would be extraneous.

Step 1: Set the Equation

The given equation is \(\sqrt{5x + 10} = -5\).

Step 2: Check Initial Condition

Since the square root of a number cannot be negative, the equation \(\sqrt{5x + 10} = -5\) will not have a real solution. This step is an early detection for possible extraneous solutions.

Step 3: Confirmation of Extraneous Solution

Even if the equation was solved further, any obtained solution will be an extraneous solution. That's because the square root of any real number is always a nonnegative number (0 or positive). Therefore, it is impossible for `\sqrt{5x + 10}` to equal any negative number such as -5.

Key Concepts

Solving Radical Equations

Solving equations involving radicals—especially square roots—is a foundational skill in algebra. When faced with a radical equation, the goal is to isolate the radical on one side and then square both sides of the equation to eliminate the radical.

However, an important initial step is to observe the properties of radicals and the numbers involved. For example, if a square root is set equal to a negative number, we can determine without further computation that there is no real solution, since square roots by definition yield nonnegative results.

Despite the lack of a real solution in such a case, going through the steps to solve can still provide valuable practice. Always remember, after squaring both sides and finding a potential solution, to substitute back into the original equation to check for extraneous solutions—these are solutions that fit the transformed equation but not the original equation.

Square Roots in Algebra

Square roots play a critical role in algebra, serving as the inverse operation of squaring a number. When you see a square root in an equation, it means you are looking for a number which, when multiplied by itself, gives the number inside the radical.

In the context of equation solving, to clear a square root, you would typically square both sides. This is because \(\sqrt{x}\)^2 = x. It’s crucial, however, to remain aware of the domain of the variable within the radical; this domain must be nonnegative in the case of real numbers.

Occasionally, an equation may have no solution if the square root is equated with a negative number, as the square root function only produces nonnegative outputs for real numbers. This understanding can save time and prevent unnecessary calculations.

No Real Solution

In algebra, we often encounter equations that lack a solution within the set of real numbers. This is typically the case when the specific conditions required for a solution do not match the properties of real numbers, such as when taking the square root of a negative value.

When such conditions arise, it's important to acknowledge that no real solution exists. The given equation \(\sqrt{5x + 10} = -5\) is a prime example. No real number 'x' can satisfy this equation because a square root cannot produce a negative result. Recognizing when an equation has no real solution is useful as it helps in focusing efforts on solvable problems and deepens understanding of the way numbers work within the system of real numbers.

Acknowledging the absence of a real solution is not synonymous with failure; instead, it is a proof of understanding the underlying principles governing the operation at hand.