Problem 71 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (4 x-1)^{2}-48=0 $$

Step-by-Step Solution

Answer

The solutions are \(\tilde x = \frac{1 + 4\sqrt{3}}{4} , x = \frac{1 - 4\sqrt{3} \4}

Step 1: Isolate the squared term

First, isolate the squared term \( (4x - 1)^2 \) by adding 48 to both sides of the equation: \[ (4x - 1)^2 - 48 + 48 = 0 + 48 \] which simplifies to \[ (4x - 1)^2 = 48 \]

Step 2: Apply the square root property

Take the square root of both sides to eliminate the square: \[ \sqrt{(4x - 1)^2} = \sqrt{48} \] which gives \[ 4x - 1 = \pm\sqrt{48} \]

Step 3: Simplify the radical

Simplify \sqrt{48} by breaking it into prime factors: \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \] So the equation becomes \[ 4x - 1 = \pm 4\sqrt{3} \]

Step 4: Solve for x

Separate the equation into two cases due to \( \pm \): \[ 4x - 1 = 4\sqrt{3} \] \[ 4x - 1 = -4\sqrt{3} \] Solve each equation by adding 1 to both sides and dividing by 4: \[4x = 4 \sqrt{3} + 1 \] \[x = \sqrt{3} + \frac{1}{4} \] \[4x = -4 \sqrt{3} + 1 \] \[x = -\frac{\frac{1}{4}}{4} \]

Step 5: Write the final solution

Combine the solutions from both cases to get the final answer: \[ x = \frac{1 + 4\frac{\ both\frac{\underline{\phantom{xx}}}{to the each}} \]

Key Concepts

square root property

To solve a quadratic equation using the square root property, you start by isolating the squared term. Once isolated, take the square root of both sides of the equation to eliminate the square.
This step reveals two possible solutions due to the property of squares: For example, if you have \( y^2 = k \), then \( y = \pm \sqrt{k} \).
This means there are always two values satisfying the equation: one positive and one negative.

isolate the variable

Isolating the variable is crucial when solving equations. In quadratic equations, this often means isolating the squared term first.
In our exercise, we isolated \( (4x - 1)^2 \) by adding 48 to both sides, transforming the equation to \( (4x - 1)^2 = 48 \). This step clears the path to apply the square root property, allowing us to eventually isolate x itself.

simplifying radicals

Simplifying radicals is an essential skill in solving quadratic equations. When simplifying \( \sqrt{48} \), break it down into its prime factors:
\[\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \].
This transformation makes it easier to work with the equation and find solutions. It's important to recognize and use properties of square roots to ensure radical expressions are in their simplest form.

solving quadratic equations

Solving quadratic equations requires understanding various techniques. In our example, we use the following steps:

• Isolate the squared term.
• Apply the square root property.
• Simplify the resulting radicals.
• Isolate the variable again.

This method can be very efficient for equations that are already set up conveniently for the square root property. Responsibilities in breaking the method down into these clear, manageable steps helps make the solution process more intuitive.