Problem 15 - Algebra 2

Question

Solve each equation. $$ 7+\sqrt{4 x+8}=9 $$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Isolate the Square Root

First, subtract 7 from both sides of the equation to isolate the square root. This gives:\[ \sqrt{4x + 8} = 9 - 7 \]Simplifying the right side, we get:\[ \sqrt{4x + 8} = 2 \]

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation:\[ (\sqrt{4x + 8})^2 = 2^2 \]This simplifies to:\[ 4x + 8 = 4 \]

Step 3: Solve for x

Subtract 8 from both sides to solve for \(x\):\[ 4x + 8 - 8 = 4 - 8 \]\[ 4x = -4 \]Now, divide both sides by 4:\[ x = \frac{-4}{4} = -1 \]

Step 4: Verify the Solution

Substitute \(x = -1\) back into the original equation to ensure it satisfies the equation:\[ 7 + \sqrt{4(-1) + 8} = 9 \]This simplifies to:\[ 7 + \sqrt{-4 + 8} = 9 \]\[ 7 + \sqrt{4} = 9 \]\[ 7 + 2 = 9 \]Since both sides equal 9, \(x = -1\) is a valid solution.

Key Concepts

Isolating the Square Root

When solving equations involving square roots, it's often necessary to isolate the square root on one side of the equation. This process simplifies the problem and prepares it for further steps. Consider the equation given: \[ 7 + \sqrt{4x + 8} = 9 \] To isolate the square root, you need to "move" other numbers to the opposite side. This is done by performing algebraic operations like addition or subtraction. Here, subtract 7 from both sides of the equation: \[ \sqrt{4x + 8} = 9 - 7 \] Simplifying gives: \[ \sqrt{4x + 8} = 2 \] By isolating the square root, you've simplified the equation to a form which is easier to manipulate in subsequent steps. Essential points to remember include ensuring that the square root is the only item remaining on one side after isolation.

Squaring Both Sides

The next step in solving equations with square roots is to eliminate the root. This is achieved by squaring both sides of the equation. In our scenario, once the square root is isolated: \[ \sqrt{4x + 8} = 2 \] You square both sides to remove the square root: \[ (\sqrt{4x + 8})^2 = 2^2 \] This makes the equation: \[ 4x + 8 = 4 \] Squaring both sides helps because the square root and the square cancel each other out, simplifying the calculations. Remember, it's critical to ensure equality by squaring the entire expressions on both sides.

Verifying Solutions

After you solve the equation, it's important to verify that your solution is correct. This ensures no mistakes were made during the solving process. For the equation solution \( x = -1 \), substitute \( x \) back into the original equation: \[ 7 + \sqrt{4(-1) + 8} = 9 \] Simplify the expression: \[ 7 + \sqrt{-4 + 8} = 9 \] \[ 7 + \sqrt{4} = 9 \] \[ 7 + 2 = 9 \] Since both sides equal 9, the original equation holds true. Verification confirms that the algebraic steps were correct. This step prevents any oversight from going unnoticed by catching calculation errors.

Algebraic Manipulation

Solving equations requires various algebraic techniques to simplify and solve for the unknown variable. In this example, several manipulations are used, such as subtracting or adding numbers and dividing terms. Let's recap: after isolating the square root and squaring both sides to obtain \[ 4x + 8 = 4 \] You needed to solve for \( x \) by eliminating constants. Subtract 8 from both sides: \[ 4x = 4 - 8 \] Which simplifies to: \[ 4x = -4 \] Finally, divide each side by 4 to solve for \( x \): \[ x = \frac{-4}{4} = -1 \] Utilizing algebraic manipulation helps in breaking down complex expressions into manageable pieces for easier solving. Mastering these techniques is crucial for efficiently solving equations across different contexts.