Problem 42 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 3 x^{2}+8=80 $$

Step-by-Step Solution

Answer

x = \pm 2\sqrt{6}

Step 1: Isolate the quadratic term

First, move the constant term to the other side by subtracting 8 from both sides of the equation: 3x^2 + 8 - 8 = 80 - 8 This simplifies to: 3x^2 = 72

Step 2: Divide by the coefficient of the quadratic term

Next, divide both sides by 3 to isolate the squared term: \( \frac{3x^2}{3} = \frac{72}{3} \) 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 \(x = \pm \sqrt{a}\): x = \pm\sqrt{24}

Step 4: Simplify the radical

Finally, simplify \( \sqrt{24} \). Notice that 24 can be factored into 4 and 6, where 4 is a perfect square: x = \pm \sqrt{4 \cdot 6} x = \pm \sqrt{4} \cdot \sqrt{6} Since \( \sqrt{4} = 2 \), the simplified form is: x = \pm 2\sqrt{6}

Key Concepts

Square Root Property

The square root property is a powerful tool in solving quadratic equations.
It helps us deal directly with quadratic terms like \(x^2\).
When you have an equation of the form \(x^2 = a\), you can apply the square root to both sides of the equation.
This gives you two possible solutions because both the positive and negative roots of the term \(\text{square root of } a\) are valid.
In mathematical terms, if \(x^2 = a\), then \(x = \pm \sqrt{a}\).
This is very handy because it allows us to break down quadratic terms without having to expand or use more complicated methods.

Simplifying Radicals

Simplifying radicals is about breaking down square roots into their simplest form.
When you have a number under the square root sign that is not a perfect square, you can often simplify it by factoring.
For example, let's say you have \( \sqrt{24} \). At first, this may look unsimplifiable, but notice that 24 can be factored into 4 and 6.
Since 4 is a perfect square (\( 4 = 2^2 \)), we can write: \( \sqrt{24} = \sqrt{4 \cdot 6} \).
Further, since \( \sqrt{4} = 2 \), we simplify this to \( 2 \sqrt{6} \).
It's important to break down the number inside the square root into its prime factors and look for pairs of numbers to simplify.

Isolating the Quadratic Term

To solve a quadratic equation, you often need to isolate the quadratic term first.
This means moving all other terms to one side of the equation so that you have your quadratic term alone on the other side.
For instance, in the equation \( 3x^2 + 8 = 80 \), we start by subtracting 8 from both sides to move the constant term:
\( 3x^2 = 72 \).
Next, you want to isolate \( x^2 \) by dividing both sides by the coefficient of \( x^2 \): \( \frac{{3x^2}}{{3}} = \frac{{72}}{{3}} \).
This simplifies to \( x^2 = 24 \).
By isolating the quadratic term, you simplify the equation, making it easier to apply the square root property and find your solution.