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Chapter 8: Problem 28

Use the square root property to solve each equation. \(3 x^{2}+8=80\)

Short Answer

Expert verified
x = \pm 2\sqrt{6}

Step by step solution

01

Isolate the quadratic term

First, subtract 8 from both sides of the equation to isolate the term involving \(x^2\): \[3x^2 + 8 - 8 = 80 - 8\] \[3x^2 = 72\]
02

Divide by the coefficient of x squared

Next, divide both sides of the equation by 3 to solve for \(x^2\): \[\frac{3x^2}{3} = \frac{72}{3}\] \[x^2 = 24\]
03

Apply the square root property

Finally, apply the square root property to both sides of the equation. Remember to include both the positive and negative roots: \[x = \pm \sqrt{24}\] \[x = \pm 2\sqrt{6}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square root property
To solve equations using the square root property, we essentially take the square root of both sides of the equation. This property is particularly useful when dealing with equations where the variable is squared. When employing the square root property, always remember to consider both the positive and negative solutions.
For example: To find the solutions to the equation \(x^2 = 24\), taking the square root of both sides gives us \(x = \sqrt{24}\) or \(x = -\sqrt{24}\).
Thus, \(x = \pm\sqrt{24}\). Simplifying further: \( \sqrt{24} = 2\sqrt{6}\), so \(x = \pm 2\sqrt{6}\).
isolating the quadratic term
Isolating the quadratic term means getting the term with the squared variable by itself on one side of the equation. This is often the first step in solving quadratics.
Consider the equation \(3x^2 + 8 = 80\). To isolate the quadratic term, we subtract 8 from both sides of the equation: \(3x^2 = 72\).
By performing this step, we make it easier to then solve for the variable, either by factoring, using the quadratic formula, or applying the square root property.
dividing by the coefficient
After isolating the quadratic term, typically the next step is to simplify the equation by dividing by the coefficient of the squared term. This allows us to have \(x^2\) by itself on one side.
For instance, after isolating \(3x^2 = 72\), we divide both sides by 3: \(x^2 = 24\).
Dividing by the coefficient simplifies the equation, making it easier to apply further methods like the square root property. Remember that whatever operation you do to one side of the equation, you must do to the other to maintain equality.

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Most popular questions from this chapter

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