Problem 13 - Algebra 2

Question

Solve each equation. $$ a^{\frac{1}{2}}+9=0 $$

Step-by-Step Solution

Answer

No real solution.

Step 1: Isolate the Power Term

To solve the equation, we first isolate the term involving the variable. We have the equation \(a^{\frac{1}{2}} + 9 = 0\). We can isolate the \(a^{\frac{1}{2}}\) by subtracting 9 from both sides: \(a^{\frac{1}{2}} = -9\).

Step 2: Analyze the Square Root

The expression \(a^{\frac{1}{2}}\) is equivalent to \(\sqrt{a}\). Since the square root of a number must be non-negative, \(\sqrt{a} = -9\) has no real solutions.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value of the unknown variable that makes the equation true. We follow a series of logical steps to isolate and solve for the variable. To do this, we often need to manipulate the equation using basic arithmetic operations.

• Isolating the variable: This involves rearranging the equation so that the variable is on one side and the constants are on the other.
• In our example equation, \(a^{\frac{1}{2}} + 9 = 0\), we want to isolate \(a^{\frac{1}{2}}\).
• We achieve this by subtracting 9 from both sides, resulting in \(a^{\frac{1}{2}} = -9\).

Solving equations often involves multiple steps and may require reversing operations. Each step should maintain the balance of the equation, meaning whatever operation you do to one side, you must do to the other.

Square Roots

Square roots are mathematical operations that "undo" squaring a number. The square root of a number \(a\) is a value \(b\) such that \(b^2 = a\). In our example, when we see \(a^{\frac{1}{2}}\), it is equivalent to \(\sqrt{a}\).

• This notation means we are looking for a number that, when squared, returns the original value \(a\).
• Importantly, the principal square root is always non-negative, as it represents the length or distance.

The properties of square roots are crucial for understanding when real solutions exist in equation problems. As you practice, remember that square roots have constraints, and this influences whether or not there is a real solution.

No Real Solutions

In mathematics, not all equations can be solved in the real numbers. An equation can be said to have "no real solutions" if there is no real number that satisfies it. This occurs because of certain characteristics of operations like square roots.

• For instance, in the equation \(a^{\frac{1}{2}} = -9\), we face such a scenario.
• The square root of a number cannot be negative, as square roots represent lengths which are inherently non-negative.
• Therefore, \(\sqrt{a} = -9\) is impossible, indicating that there are no real solutions to this equation.

Instead, we might explore complex numbers to find solutions, but in the realm of real numbers, some equations remain unsolvable. Understanding these limitations helps in identifying when a different approach might be needed.