Problem 417 - Elementary Algebra

Question

In the following exercises, solve. $$ \sqrt{4 v+3}=\sqrt{v-6} $$

Step-by-Step Solution

Answer

The solution is \(v = -3\).

Step 1 - Equalize the Radicands

Since the square roots on both sides of the equation are equal, we can set the expressions inside the square roots equal to each other. So, we have: \[ 4v + 3 = v - 6 \]

Step 2 - Isolate Variable Terms

Subtract \(v\) from both sides to start isolating the variable \(v\). This gives: \[ 4v - v + 3 = -6 \] Simplifying, we get: \[ 3v + 3 = -6 \]

Step 3 - Solve for \(v\)

Subtract 3 from both sides to further isolate \(v\): \[ 3v + 3 - 3 = -6 - 3 \] This simplifies to: \[ 3v = -9 \] Finally, divide both sides by 3 to solve for \(v\): \[ v = \frac{-9}{3} = -3 \]

Step 4 - Check the Solution

Substitute \(v = -3\) back into the original equation to ensure it works: \[ \sqrt{4(-3) + 3} = \sqrt{(-3) - 6} \] Simplify inside the square roots: \[ \sqrt{-12 + 3} = \sqrt{-9} \] Which simplifies to: \[ \sqrt{-9} = \sqrt{-9} \] Since both sides are equal, \(v = -3\) is a valid solution.

Key Concepts

Square Roots

Square roots are mathematical operations that return a number whose square gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \). When solving equations involving square roots, it’s helpful to equalize the expressions inside the square roots, known as the radicands. In the given exercise, the equation \( \sqrt{4v + 3} = \sqrt{v - 6} \) requires us to set 4v + 3 equal to v - 6 to eliminate the square roots and simplify the equation. This step ensures we can work with a simpler, algebraic form.

Isolating Variables

Isolating the variable means moving all instances of that variable to one side of the equation. This process makes it easier to solve for that variable. Here, we start with \( 4v + 3 = v - 6 \). To isolate v, we subtract v from both sides, resulting in \( 3v + 3 = -6 \). Then, we subtract 3 from both sides to get \( 3v = -9 \). Finally, we divide by 3, giving us \( v = -3 \). Each step simplifies the equation until the variable is isolated and solved.

Checking Solutions

Checking the solution is a crucial step to ensure that our value for the variable satisfies the original equation. In our example, we substitute \( v = -3 \) back into the original equation: \( \sqrt{4(-3) + 3} = \sqrt{(-3) - 6} \). Simplification gives \( \sqrt{-12 + 3} = \sqrt{-9} \), resulting in \( \sqrt{-9} = \sqrt{-9} \). Since both sides are equal, v = -3 is indeed a valid solution. Always check your solutions as sometimes, due to extraneous roots introduced through squaring, not all solutions may be valid for the original equation.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process helps in making the equations easier to solve. For instance, when you have \( 3v + 3 = -6 \, you subtract 3 from both sides to simplify the expression to \ 3v = -9 \. Further simplification by dividing both sides by 3 gives \ v = -3 \). Simplifying each part of your equation step by step reduces complexity and aids in finding the correct solution more efficiently.