Problem 36 $-8 p^{2}=-32$... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 9: Problem 36

$-8 p^{2}=-32$

Short Answer

Expert verified

The solutions for $p$ are $2$ and $-2$.

Step by step solution

Isolate the quadratic term

Divide both sides of the equation by (-8) to isolate the quadratic term. This gives: \[-8 p^{2} = -32\]\[p^{2} = \frac{-32}{-8}\]\[p^{2} = 4\]

Take the square root of both sides

To solve for $p$, take the square root of both sides of the equation. This results in: \[p = \pm \sqrt{4}\]\[p = \pm 2\]

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Key Concepts

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

Isolating the Variable

To solve a quadratic equation, it is crucial to first isolate the variable. In our example, the original equation is $-8 p^{2}=-32$. The goal here is to have only the term with the variable on one side of the equation. We do this by dividing both sides by the same number, which in this case is $-8$. This operation simplifies the equation to: \[p^{2} = \frac{-32}{-8}\] and then further to: \[p^{2} = 4\].
Now, we have successfully isolated the quadratic term $p^2$, meaning there are no other numbers or variables on its side of the equation.
This simplification is the first and an important step towards solving the quadratic equation.

Taking the Square Root

Once the variable term is isolated, the next step is to solve for the variable by taking the square root of both sides of the equation. From our isolated equation $p^2 = 4$, taking the square root gives us: \[p = \pm \sqrt{4}\].
Here, it's important to remember the square root property which tells us that for any positive number $a$, $\sqrt{a}$ and $\sqrt{-a}$ are solutions.
So, $\sqrt{4}$ results in $\pm2$.
Taking the square root helps in 'undoing' the square on the variable, essentially solving for the variable.

Positive and Negative Solutions

Quadratic equations often have two solutions due to the nature of squaring a number, which always results in a positive value. When the square root is taken, it introduces both positive and negative solutions.
In our example, we have: \[p = \pm 2\].
This indicates two solutions: \p = 2\ and \p = -2\.
These are called the roots of the equation, and they show that both $2^2=4$ and $(-2)^2=4$ hold true.
Understanding the positive and negative solutions is critical as it fully solves the quadratic equation, providing all possible values for the variable.

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91影视

\(-8 p^{2}=-32\)

Short Answer

Step by step solution

Isolate the quadratic term

Take the square root of both sides

Key Concepts

Isolating the Variable

Taking the Square Root

Positive and Negative Solutions

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