Problem 49 - Elementary Algebra

Question

Solve using the Square Root Property. \(6 c^{2}+4=29\)

Step-by-Step Solution

Answer

c = \pm \frac{5\sqrt{6}}{6}

Step 1: Isolate the Quadratic Term

Start by isolating the quadratic term on one side of the equation. Subtract 4 from both sides to get: \[ 6c^2 + 4 - 4 = 29 - 4 \]\[ 6c^2 = 25 \]

Step 2: Divide by the Coefficient of the Quadratic Term

Divide both sides by the coefficient of the quadratic term, which is 6, to simplify the equation:\[ \frac{6c^2}{6} = \frac{25}{6} \]\[ c^2 = \frac{25}{6} \]

Step 3: Apply Square Root Property

Apply the Square Root Property to both sides of the equation to solve for c:\[ c = \pm \sqrt{\frac{25}{6}} \]

Step 4: Simplify the Square Root

Simplify the square root to get the final solution. We know that the square root of a fraction is the square root of the numerator divided by the square root of the denominator:\[ c = \pm \frac{\sqrt{25}}{\sqrt{6}} \]Since \(\sqrt{25} = 5\), the solution is:\[ c = \pm \frac{5}{\sqrt{6}} \]Rationalize the denominator to get:\[ c = \pm \frac{5\sqrt{6}}{6} \]

Key Concepts

solving quadratic equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). In this exercise, we have an equation \(6c^2 + 4 = 29\).

The goal is to isolate the squared term. This process involves:

• Moving all other terms to one side.
• Simplifying the equation to make it easier to solve.

For instance, subtract 4 on both sides to get \(6c^2 = 25\). After isolating the quadratic term, we can solve for \(c\).

square root property

The Square Root Property is a method used to solve quadratic equations that do not have a linear term. Once the quadratic term is isolated and simplified, we apply the Square Root Property: if \(x^2 = k\), then \(x = \pm\sqrt{k}\).

For our equation, after isolating and simplifying, we get \(c^2 = \frac{25}{6}\). Applying the Square Root Property gives us two possible solutions: \(c = \pm\sqrt{\frac{25}{6}}\).

Always remember to include both the positive and negative roots.

simplifying radicals

Simplifying radicals means writing them in their simplest form. For our equation, we have \(c = \pm\sqrt{\frac{25}{6}}\).

To simplify, we take the square root of both numerator and denominator: \( c = \pm\frac{\sqrt{25}}{\sqrt{6}}\). Since \(\sqrt{25} = 5\), the equation simplifies to \( c = \pm\frac{5}{\sqrt{6}}\).

Use this method to break down complex radicals, making them easier to work with.

rationalizing the denominator

Rationalizing the denominator is the process of removing radicals from the denominator of a fraction. Our simplified solution is \(c = \pm\frac{5}{\sqrt{6}}\).

To rationalize, we multiply both numerator and denominator by the radical in the denominator:
\(c = \pm\frac{5}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{6}} = \pm\frac{5\sqrt{6}}{6}\).

This process gives us a fraction with a rational number in the denominator, which is generally preferred in mathematics.