Problem 31 - Intermediate Algebra

Question

Solve. \(\sqrt{4 x-3}=7\)

Step-by-Step Solution

Answer

The solution is \( x = 13 \).

Step 1: Isolate the Square Root

The given equation is \( \sqrt{4x - 3} = 7 \). The square root is already isolated on one side of the equation, so we can proceed to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \( (\sqrt{4x - 3})^2 = 7^2 \). This simplifies to \( 4x - 3 = 49 \).

Step 3: Solve for x

Add 3 to both sides of the equation to get \( 4x = 52 \). Then, divide both sides by 4 to find \( x = \frac{52}{4} = 13 \).

Step 4: Verify the Solution

Substitute \( x = 13 \) back into the original equation to check: \( \sqrt{4(13) - 3} = \sqrt{52 - 3} = \sqrt{49} = 7 \). Since both sides are equal, \( x = 13 \) is the correct solution.

Key Concepts

Square Roots

Square roots are mathematical tools used to find a number which, when multiplied by itself, gives the original number. In equations like \( \sqrt{4x - 3} = 7 \), the expression under the square root (or radicand) represents a value which, once rooted, equals the number on the other side of the equation. Understanding square roots is crucial when dealing with radical equations.

Radicals, especially square roots, often appear in equations that need solving by isolating or manipulating expressions, whether for physics problems, geometry, or algebra itself. Recognizing that a square root can be reversed by squaring—and thus simplifying equations—is a key skill. This technique helps remove radicals and allows you to work with linear or polynomial expressions more straightforwardly.

Isolating Variables

Isolating a variable is a fundamental step in solving equations. It involves rearranging the equation so that the variable you're solving for is on one side by itself. In our equation, \( \sqrt{4x - 3} = 7 \), the square root expression \( \sqrt{4x - 3} \) is already isolated. This means we can efficiently move to the step of squaring both sides without additional adjustments.

Isolating the variable often involves using various operations to simplify or remove other terms on the same side as the variable. This could mean adding, subtracting, multiplying, or dividing parts of the equation. The goal is always to have the variable by itself, which simplifies the calculation and helps clearly visualize the problem at hand.

Squaring Both Sides

Squaring both sides of an equation is a common tactic used when solving equations with a square root. In the equation \( \sqrt{4x-3} = 7 \), squaring both sides allows the equation to transform from a radical expression into a simple algebraic expression. This step involve raising each side of the equation to the power of 2, thereby eliminating the square root and yielding a solvable linear equation.

Here's how it works:

• We square the left side: \((\sqrt{4x - 3})^2 = 4x - 3\)
• We also square the right side: \(7^2 = 49\)
This gives us the new equation \(4x - 3 = 49\). Now, you can solve this equation through basic arithmetic steps. Squaring both sides is an elegant method to transform and simplify equations, but remember, it's pivotal to verify the solution since squaring can sometimes introduce extraneous solutions.

Verifying Solutions

Once you find a solution to an equation, verifying it is an essential step to ensure accuracy and avoid mistakes. After solving \(4x - 3 = 49\), we determined that \(x = 13\). Substitute this value back into the original radical equation \( \sqrt{4x - 3} = 7 \), to confirm it satisfies the equation.

Substitution shows:

• Plug \(x = 13\) into \( \sqrt{4(13) - 3} \)
• Calculate \(4(13) - 3 = 49\)
• Check if \( \sqrt{49} = 7 \)

Since both sides equal, the solution is verified. This process of verification not only confirms the correctness but also rules out any extraneous solutions arising from squaring both sides.