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Chapter 8: Problem 29

Use the square root property to solve each equation. \(3 x^{2}-10=86\)

Short Answer

Expert verified
The solutions are \( x = 4\sqrt{2} \) and \( x = -4\sqrt{2} \).

Step by step solution

01

- Isolate the Quadratic Term

First, add 10 to both sides of the equation to isolate the quadratic term. The equation becomes: \[ 3x^2 - 10 + 10 = 86 + 10 \] Simplifying, we get: \[ 3x^2 = 96 \]
02

- Divide by the Coefficient of the Quadratic Term

Next, divide both sides of the equation by 3 to solve for \( x^2 \). So, \[ \frac{3x^2}{3} = \frac{96}{3} \] This simplifies to: \[ x^2 = 32 \]
03

- Apply the Square Root Property

Now, apply the square root property to solve for \( x \). The square root property states that if \( x^2 = a \), then \( x = \pm \sqrt{a} \). Therefore, \[ x = \pm \sqrt{32} \] Simplifying, we get: \[ x = \pm 4\sqrt{2} \]

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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 quadratic equations, one useful tool is the square root property. This property states that if you have an equation where the quadratic term equals a number, you can find the solutions by taking the square root of both sides. Specifically, if you have an equation of the form \(x^2 = a\), then the solutions are \(x = \pm \sqrt{a}\). This gives both the positive and negative roots since squaring either a positive or a negative number gives the same result.
Isolating Quadratic Term
To apply the square root property effectively, the equation must first be in a form where the quadratic term is isolated on one side of the equation. In the exercise, we start with the equation \(3x^2 - 10 = 86\). To isolate the quadratic term \(3x^2\), we need to get rid of the constant term on the left side by adding 10 to both sides. This gives us \(3x^2 = 96\). Breaking down the equation step by step helps us better manage these transformations.
Simplifying Equations
Once the quadratic term is isolated, the next step is to simplify the equation further. This typically involves handling any coefficients of the quadratic term. In our exercise, after isolating \(3x^2\), we still have a coefficient of 3. Simplifying means dividing both sides of the equation by 3, which gives us \(x^2 = 32\). Each step in simplifying helps make the problem easier to handle, leading us closer to applying the square root property.
Quadratic Term Coefficient
The coefficient of a quadratic term can often complicate the process of solving equations. In our exercise, the coefficient is 3. To proceed, we divided the entire equation by this coefficient. This resulted in the simpler equation \(x^2 = 32\). Recognizing when to divide by the coefficient is key to getting a clear path to the solution. Finally, we apply the square root property to find the solutions as \(x = \pm 4 \sqrt{2}\).

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