Problem 33 Solve equation by using the squa... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 9: Problem 33

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

Short Answer

Expert verified

x = \pm 2\sqrt{2}

Step by step solution

- Isolate the quadratic term

First, divide both sides of the equation by (-8) to isolate the quadratic term. $ \frac{-8x^{2}}{-8}=\frac{-64}{-8}$ which simplifies to $ x^{2}=8 $

- Apply the square root property

Next, apply the square root property which states that if $x^2 = a$, then $x = \pm \sqrt{a}$. Using this property, we get: $ x = \pm \sqrt{8} $

- Simplify the radical

Simplify the radical $ \sqrt{8} $. Since $8 = 4 \cdot 2 $ and $4$ is a perfect square:$ x = \pm \sqrt{4 \cdot 2} = \pm \sqrt{4} \cdot \sqrt{2} = \pm 2\sqrt{2}$

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Key Concepts

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

Isolate Quadratic Term

To solve a quadratic equation using the square root property, the first step is to isolate the quadratic term. In other words, we need to get the term with the variable squared by itself on one side of the equation. For example, in the equation $-8 x^{2}=-64$, the quadratic term is $x^2$.
To isolate $x^2$, we divide both sides by $-8$:
\[ \frac{-8x^{2}}{-8}=\frac{-64}{-8} \]
This simplifies to: \[ x^{2}=8 \]
By isolating the quadratic term, we prepare the equation for the next step, which is applying the square root property.

Square Root Property

Once the quadratic term is isolated, we use the square root property. This property states that if $x^2 = a$, then $x = \pm \sqrt{a}$.
Let's apply this property to our equation $x^2 = 8$:
\[ x = \pm \sqrt{8} \]
This means that $x$ can be either the positive or negative square root of 8.
Remember, taking the square root of both sides introduces both the positive and negative solutions.

Simplifying Radicals

Simplifying radicals involves breaking down the number inside the square root to its simplest form. To simplify $ \sqrt{8} $, we recognize that 8 can be written as 4 multiplied by 2:
\[ \sqrt{8} = \sqrt{4 \times 2} \]
Since 4 is a perfect square, we simplify further:
\[ \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2} \]
Therefore, the solutions to the equation $-8 x^{2}=-64$ are:
\[ x = \pm 2 \sqrt{2} \]
Ensuring our radicals are simplified makes the final answer clearer and more precise.

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91影视

Solve equation by using the square root property. Simplify all radicals. \(-8 x^{2}=-64\)

Short Answer

Step by step solution

- Isolate the quadratic term

- Apply the square root property

- Simplify the radical

Key Concepts

Isolate Quadratic Term

Square Root Property

Simplifying Radicals

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