Problem 43 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 3 x^{2}-8=64 $$

Step-by-Step Solution

Answer

x = \pm 2\sqrt{6}

Step 1: Isolate the quadratic term

Begin by isolating the term that contains the variable. To do this, add 8 to both sides of the equation: \[3x^{2} - 8 + 8 = 64 + 8\]This simplifies to: \[3x^{2} = 72\]

Step 2: Divide by the coefficient of the quadratic term

Next, divide both sides by 3 to solve for \(x^2\): \[\frac{3x^{2}}{3} = \frac{72}{3}\]This simplifies to: \[x^{2} = 24\]

Step 3: Apply the square root property

Now, apply the square root property, which states that if \(x^{2} = a\), then \(x = \pm \sqrt{a}\). So, take the square root of both sides of the equation: \[x = \pm \sqrt{24}\]

Step 4: Simplify the radical

Finally, simplify \(\sqrt{24}\). The prime factorization of 24 is \[24 = 2^3 \times 3\]Recognize that \(\sqrt{24}\) can be rewritten using the property \(\sqrt{a\times b} = \sqrt{a} \times \sqrt{b}\). Therefore: \[\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\]Thus, the solution is: \[x = \pm 2\sqrt{6}\]

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be written in the form of \[ax^{2} + bx + c = 0\], where 'a', 'b', and 'c' are constants, and 'x' represents an unknown variable. In our problem, the quadratic equation is \[3x^{2} - 8 = 64\]. Let's dive deeply into the specifics of each term and how to solve such equations.
Normally, when we solve quadratic equations, we look for the values of 'x' that satisfy the equation.

Isolating Terms

Isolating terms is a crucial step in solving equations. To isolate a term means to manipulate the equation until the term containing the variable stands alone on one side of the equation.
In our problem, we start with \[3x^{2} - 8 = 64\]. To isolate the term involving \(x\), we add 8 to both sides, resulting in \[3x^{2} = 72\].
By carefully performing these operations, we simplify the equation and make it easier to solve for x.

Simplifying Radicals

Simplifying radicals involves breaking down a square root into its simplest form. Radicals can often be simplified by recognizing that certain factors of the number under the square root can be squared.
For example, with \[x = \pm \sqrt{24}\], we can simplify the radical by writing \[24 = 4 \times 6\]. The square root of 4 is 2, so our expression becomes \[\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\].
This process makes the radical simpler and the solution clearer.

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
In our problem, we need the prime factors of 24, which are \[2^{3} \times 3\]. Knowing this helps us simplify the square root more effectively.
Prime factorization is useful in various areas of algebra and arithmetic, including simplifying radicals as demonstrated here.