Problem 53 - Algebra 2

Question

For what values of \(a\) and \(b\) does \(\sqrt{a}+\sqrt{b}=\sqrt{a+b} ?\)

Step-by-Step Solution

Answer

The values of a and b that satisfy the equation are \(a=0, b=any real number\) or \(a=any real number, b=0\).

Step 1: Squaring the equation

To simplify, square both sides of the equation \(\sqrt{a} + \sqrt{b} = \sqrt{a+b}\). This results in \(a + 2\sqrt{a}\sqrt{b} + b = a + b\).

Step 2: Simplifying equation

Next, cancel out \(a+b\) on both sides of the equation - thus getting \(2\sqrt{a}\sqrt{b} = 0\).

Step 3: Solving for \(a\) and \(b\)

Finally, either \(\sqrt{a}=0\) or \(\sqrt{b}=0\), which implies that either \(a=0\) or \(b=0\). Therefore, the values of \(a\) and \(b\) that satisfy the equation are \(a=0, b=any real number\) or \(a=any real number, b=0\).

Key Concepts

Square Roots

Square roots are an essential concept in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, both 3 and -3 are square roots of 9 because

• \(3 \times 3 = 9\)
• \(-3 \times -3 = 9\)

In the context of equations involving square roots, we often see structures like \(\sqrt{a}\), which demands careful manipulation.
When manipulating square roots, it’s crucial to remember they are always non-negative in real numbers, unless explicitly working with complex numbers.
Knowing how to simplify square root expressions is vital for solving radical equations efficiently.
This usually involves identifying perfect squares and simplifying the expression by removing square roots when possible.

Equation Solving

Solving equations involves finding the unknown variables that make an equation true. In radical equations, such as the one given in the exercise \(\sqrt{a} + \sqrt{b} = \sqrt{a+b}\), the goal is to isolate the variables.
By squaring both sides of the equation, you remove the radicals, making the equation easier to solve. Squaring effectively cancels out the square root, leaving linear terms you can solve for.
However, be cautious when squaring because it can sometimes introduce extraneous solutions, which are solutions that don't satisfy the original equation.
Thus, it's important to verify all solutions in the original equation.

Simplification

Simplification is a crucial step in solving equations that include square roots. It involves reducing an expression or equation to its simplest form. In the given solution, after squaring both sides of the equation \(\sqrt{a}+\sqrt{b}=\sqrt{a+b}\), you are left with
\[a + 2\sqrt{a}\sqrt{b} + b = a + b\]
Here, you simplify by canceling \(a + b\) from both sides, helping you focus on the remaining term.
This leaves you with \(2\sqrt{a}\sqrt{b} = 0\). Simplification is about recognizing relationships among terms and reductions that don't change the value of the expression but make it easier to manage.
The goal is to make the equation as straightforward as possible for solving the unknowns.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding how to work with these expressions is key to handling complex equations.
In this exercise, the expression \(\sqrt{a} + \sqrt{b}\) is combined with \(\sqrt{a+b}\) to form a more complex structure.
When similar terms or expressions are present, they can often be combined. For instance, in the simplification step, a knowledge of algebraic properties allows you to recognize that some terms cancel out.
Further, defining the expression correctly is crucial for accurate manipulation.
Understanding the rules of operations, such as how to handle radicals and distribute terms across equations, is fundamental in orchestrating the simplification and solving of radical equations. Remember, every algebraic manipulation rests on these core rules.