Question

Solve: \(2 \sqrt{x-3}+5=8\)

Short answer

x = \[ \frac{21}{4} \]

Step-by-step solution

- Isolate the square root term

Start by isolating the square root term on one side of the equation. Subtract 5 from both sides: \[2 \sqrt{x-3} + 5 - 5 = 8 - 5\]This simplifies to: \[2 \sqrt{x-3} = 3\]

- Divide both sides by 2

Next, divide both sides by 2 to completely isolate the square root term: \[\frac{2 \sqrt{x-3}}{2} = \frac{3}{2}\]This simplifies to: \[\sqrt{x-3} = \frac{3}{2}\]

- Eliminate the square root

Square both sides of the equation to eliminate the square root: \[\big(\sqrt{x-3}\big)^2 = \big(\frac{3}{2}\big)^2\]This gives: \[x - 3 = \frac{9}{4}\]

- Solve for x

Add 3 to both sides to solve for x: \[x - 3 + 3 = \frac{9}{4} + 3\]Convert 3 to a fraction with the same denominator: \[3 = \frac{12}{4}\]So, \[x = \frac{9}{4} + \frac{12}{4} = \frac{21}{4}\]

Key concepts

Isolating the Square Root Term

Before solving a radical equation, it's crucial to isolate the square root term. Start by moving all other terms to the opposite side of the equation. In the given problem, you have the equation:
Start by isolating the square root term on one side of the equation. Subtract 5 from both sides: 2 Isolating the square root term is easy once you understand the basic idea: remove everything else from the side with the square root. This can be achieved by:

• Subtracting or adding numbers and variables
• Dividing or multiplying if necessary

Always keep your equation balanced by performing operations on both sides.

Eliminating the Square Root

After isolating the square root, the next step is to eliminate it to solve for the variable inside. You do this by squaring both sides of the equation.

For our example:
To eliminate the square root:

Since squaring is the inverse operation of taking a square root, squaring will cancel out the square root.

Remember, after this step, you no longer have a square root term in your equation—all that's left are algebraic terms.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate the variable we're solving for. Once the square root is eliminated, you're left with a simpler equation:

First, write down the equation and identify which algebraic operations are needed—addition, subtraction, multiplication, or division.

In this specific problem, we needed to:

• Convert a whole number to a fraction
• Combine fractions

It's important to keep track of all sides of the equation and to perform each operation accurately to ensure correctness.

Fraction Addition

When solving an equation, you may need to work with fractions. In this example, we converted the number 3 to a fraction to match the denominator of 9/4, so we could add them together.

To add fractions, ensure they have a common denominator. Once both fractions are in the same form, simply add the numerators:

So: ! To properly add the fractions, follow these steps:

• Convert whole numbers to fractions with the same denominator
• Add the numerators
• Keep the same denominator

Finally, simplify the resulting fraction if necessary.